Lecture Notes — Probability Theory
Manuel Cabral Morais
Department of Mathematics
Instituto Superior Técnico
Lisbon, September 2009/10 — January 2010/11
Contents
0. Warm up
0.1 Historical note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.2 (Symmetric) random walk . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Probability spaces
1.1 Random experiments . . . . . . . . . . . . . . .
1.2 Events and classes of sets . . . . . . . . . . . . .
1.3 Probabilities and probability functions . . . . .
1.4 Distribution functions; discrete, absolutely
probabilities . . . . . . . . . . . . . . . . . . . .
1.5 Conditional probability . . . . . . . . . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
continuous
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and
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. . .
2 Random variables
2.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Combining random variables . . . . . . . . . . . . . . . . . . . .
2.3 Distributions and distribution functions . . . . . . . . . . . . . .
2.4 Key r.v. and random vectors and distributions . . . . . . . . . .
2.4.1 Discrete r.v. and random vectors . . . . . . . . . . . . .
2.4.2 Absolutely continuous r.v. and random vectors . . . . . .
2.5 Transformation theory . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Transformations of r.v., general case . . . . . . . . . . .
2.5.2 Transformations of discrete r.v. . . . . . . . . . . . . . .
2.5.3 Transformations of absolutely continuous r.v. . . . . . .
2.5.4 Transformations of random vectors, general case . . . . .
2.5.5 Transformations of discrete random vectors . . . . . . . .
2.5.6 Transformations of absolutely continuous random vectors
2.5.7 Random variables with prescribed distributions . . . . .
ii
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1
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11
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mixed
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55
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104
3 Independence
3.1 Fundamentals . . . . . . . . . . . . . .
3.2 Independent r.v. . . . . . . . . . . . . .
3.3 Functions of independent r.v. . . . . .
3.4 Order statistics . . . . . . . . . . . . .
3.5 Constructing independent r.v. . . . . .
3.6 Bernoulli process . . . . . . . . . . . .
3.7 Poisson process . . . . . . . . . . . . .
3.8 Generalizations of the Poisson process .
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4 Expectation
4.1 Definition and fundamental properties . . . . . . .
4.1.1 Simple r.v. . . . . . . . . . . . . . . . . . . .
4.1.2 Non negative r.v. . . . . . . . . . . . . . . .
4.1.3 Integrable r.v. . . . . . . . . . . . . . . . . .
4.1.4 Complex r.v. . . . . . . . . . . . . . . . . .
4.2 Integrals with respect to distribution functions . . .
4.2.1 On integration . . . . . . . . . . . . . . . .
4.2.2 Generalities . . . . . . . . . . . . . . . . . .
4.2.3 Discrete distribution functions . . . . . . . .
4.2.4 Absolutely continuous distribution functions
4.2.5 Mixed distribution functions . . . . . . . . .
4.3 Computation of expectations . . . . . . . . . . . . .
4.3.1 Non negative r.v. . . . . . . . . . . . . . . .
4.3.2 Integrable r.v. . . . . . . . . . . . . . . . . .
4.3.3 Mixed r.v. . . . . . . . . . . . . . . . . . . .
4.3.4 Functions of r.v. . . . . . . . . . . . . . . . .
4.3.5 Functions of random vectors . . . . . . . . .
4.3.6 Functions of independent r.v. . . . . . . . .
4.3.7 Sum of independent r.v. . . . . . . . . . . .
4.4 Lp spaces . . . . . . . . . . . . . . . . . . . . . . .
4.5 Key inequalities . . . . . . . . . . . . . . . . . . . .
4.5.1 Young’s inequality . . . . . . . . . . . . . .
4.5.2 Hölder’s moment inequality . . . . . . . . .
4.5.3 Cauchy-Schwarz’s moment inequality . . . .
4.5.4 Lyapunov’s moment inequality . . . . . . . .
iii
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110
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. 141
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145
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. 179
4.6
4.5.5 Minkowski’s moment inequality .
4.5.6 Jensen’s moment inequality . . .
4.5.7 Chebyshev’s inequality . . . . . .
Moments . . . . . . . . . . . . . . . . . .
4.6.1 Moments of r.v. . . . . . . . . . .
4.6.2 Variance and standard deviation .
4.6.3 Skewness and kurtosis . . . . . .
4.6.4 Covariance . . . . . . . . . . . . .
4.6.5 Correlation . . . . . . . . . . . .
4.6.6 Moments of random vectors . . .
4.6.7 Multivariate normal distributions
4.6.8 Multinomial distributions . . . .
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5 Convergence of sequences of random variables
5.1 Modes of convergence . . . . . . . . . . . . . . . . . . .
5.1.1 Convergence of r.v. as functions on Ω . . . . . .
5.1.2 Convergence in distribution . . . . . . . . . . .
5.1.3 Alternative criteria . . . . . . . . . . . . . . . .
5.2 Relationships among the modes of convergence . . . . .
5.2.1 Implications always valid . . . . . . . . . . . . .
5.2.2 Counterexamples . . . . . . . . . . . . . . . . .
5.2.3 Implications of restricted validity . . . . . . . .
5.3 Convergence under transformations . . . . . . . . . . .
5.3.1 Continuous mappings . . . . . . . . . . . . . . .
5.3.2 Algebraic operations . . . . . . . . . . . . . . .
5.4 Convergence of random vectors . . . . . . . . . . . . .
5.5 Limit theorems for Bernoulli summands . . . . . . . .
5.5.1 Laws of large numbers for Bernoulli summands
5.5.2 Central limit theorems for Bernoulli summands
5.5.3 The Poisson limit theorem . . . . . . . . . . . .
5.6 Weak law of large numbers . . . . . . . . . . . . . . . .
5.7 Strong law of large numbers . . . . . . . . . . . . . . .
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Warm up
0.1
Historical note
Mathematical probability has its origins in games of chance [...]. Early
calculations involving dice were included in a well-know and widely distributed
poem entitled De Vetula.1 Dice and cards continued as the main vessels of
gambling in the 15th. and 16th. centuries [...]. [...] (G. Cardano) went so far
as to write a book, On games of chance, sometime shortly after 1550. This was
not published however until 1663, by which time probability theory had already
had its official inauguration elsewhere.
It was around 1654 that B. Pascal and P. de Fermat generated a celebrated
correspondence about their solutions of the problem of the points. These were
soon widely known, and C. Huygens developed these ideas in a book published
in 1657, in Latin. [...] the intuitive notions underlying this work were similar
to those commonly in force nowdays.
These first simple ideas were soon extended by Jacob Bernoulli in Ars
conjectandi (1713) and by A. de Moivre in Doctrine of chances (1718, 1738,
1756). [...]. Methods, results, and ideas were all greatly refined and generalized
by P. Laplace [...]. Many other eminent mathematicians of this period wrote
on probability: Euler, Gauss, Lagrange, Legendre, Poisson, and so on.
However, as ever harder problems were tackled by ever more powerful
mathematical techniques during the 19th. century, the lack of a well-defined
axiomatic structure was recognized as a serious handicap. [...] A. Kolmogorov
provided the axioms which today underpin most mathematical probability.
Grimmett and Stirzaker (2001, p. 571)
1
De vetula (”The Old Woman”) is a long thirteenth-century poem written in Latin. (For more details
see http://en.wikipedia.org/wiki/De vetula.)
1
For more extensive and exciting accounts on the history of Statistics and Probability,
we recommend:
• Hald, A. (1998). A History of Mathematical Statistics from 1750 to 1930. John
Wiley & Sons. (QA273-280/2.HAL.50129);
• Stigler, S.M. (1986). The History of Statistics: the Measurement of Uncertainty
Before 1900. Belknap Press of Harvard University Press. (QA273-280/2.STI.39095).
0.2
(Symmetric) random walk
This section is inspired by Karr (1993, pp. 1–14) and has the sole purpose of:
• illustrating concepts such as probability, random variables, independence,
expectation and convergence of random sequences, and recall some limit theorems;
• drawing our attention to the fact that exploiting the special structure of a random
process can provide answers for some of the questions raised.
Informal definition 0.1 — Symmetric random walk
The symmetric random walk (SRW) is a random experiment which can result from the
observation of a particle moving randomly on Z = {. . . , −1, 0, 1, . . .}. Moreover, the
particle starts at the origin at time 0, and then moves either one step up or one step down
with equal likelihood.
•
Remark 0.2 — Applications of random walk
The path followed by atom in a gas moving under the influence of collisions with other
atoms can be described by a random walk (RW). Random walk has also been applied in
other areas such as:
• economics (RW used to model shares prices and other factors);
• population genetics (RW describes the statistical properties of genetic drift);2
2
Genetic drift is one of several evolutionary processes which lead to changes in allele frequencies over
time.
2
• mathematical ecology (RW used to describe individual animal movements, to
empirically support processes of biodiffusion, and occasionally to model population
dynamics);
• computer science (RW used to estimate the size of the Web).3
•
The next proposition provides answers to the following questions:
• How can we model and analize the symmetric random walk?
• What random variables can arise from this random experiment and how can we
describe them?
Proposition 0.3 — Symmetric random walk (Karr, 1993, pp. 1–4)
1. The model
Let:
• ωn be the step at time n (ωn = ±1);
• ω = (ω1 , ω2 , . . .) be a realization of the random walk;
• Ω be the sample space of the random experiment, i.e. the set of all possible
realizations.
2. Random variables
Two random variables immediately arise:
• Yn defined as Yn (ω) = ωn , the size of the nth. step;4
• Xn which represents
P
Xn (ω) = ni=1 Yi (ω).
the
position
at
time
n
and
is
defined
as
3. Probability and independence
The sets of outcomes of this random experiment are termed events. An event A ⊂ Ω
occurs with probability P (A).
Recall that a probability function is countable additive, i.e. for sequences of
(pairwise) disjoint events A1 , A2 , . . . (Ai ∩ Aj = ∅, i 6= j) we have
3
4
For more applications check http://en.wikipedia.org/wiki/Random walk.
Steps are functions defined on the sample space. Thus, steps are random variables.
3
P
+∞
[
!
Ai
=
i=1
+∞
X
P (Ai ).
(1)
i=1
Invoking the random and symmetric character of this walk, and assuming that
the steps are independent and identically distributed, all 2n possible values of
(Y1 , . . . , Yn ) are equally likely, and, for every (y1 , . . . , yn ) ∈ {−1, 1}n ,
P (Y1 = y1 , . . . , Yn = yn ) =
n
Y
P (Yi = yi )
(2)
i=1
n
1
=
.
2
(3)
The interpretation of (2) is absence of probabilistic interaction or independence.
4. First calculations
Let us assume from now on that X0 = 0. Then:
• |Xn | ≤ n, ∀n ∈ IN ;
• Xn is even at even times (n mod 2 = 0) (e.g. X2 cannot be equal to 1);
• Xn is odd at odd times (n mod 2 = 1) (e.g. X1 cannot be equal to 0).
is an integer (n mod 2 = k mod 2) then
If n ∈ IN , k ∈ {−n, . . . , 0, . . . , n}, and n+k
2
n+k
the event {Xn = k} occurs if 2 of the steps Y1 , . . . , Yn are equal to 1 and the
remainder are equal to −1.
In fact, Xn = k if we observe a steps up and b steps down where


 a, b ∈ {0, . . . , n}
(a, b) :
a+b=n


a−b=k
that is, a =
n+k
2
(4)
and a has to be an integer in {0, . . . , n}.
As a consequence,
P (Xn = k) =
n
n+k
2
n
1
×
,
2
(5)
for n ∈ IN , k ∈ {−n, . . . , 0, . . . , n}, n+k
∈ {0, 1, . . . , n}. Recall that the binomial
2
n
coefficient n+k (it often reads as “n choose n+k
”) represents the number of
2
2
size− n+k
subsets of a set of size n.
2
4
More generally,
P (Xn ∈ B) = P ({ω : Xn (ω) ∈ B})
X
=
P (Xn = k),
(6)
k∈B∩{−n,...,0,...,n}
for any real set B ⊂ IR, by countable additivity of the probability function P .
(Rewrite (6) taking into account that n and k have to be both even or both odd.) •
Remark 0.4 — Further properties of the symmetric random walk (Karr, 1993,
pp. 4–5)
Exploiting the special structure of the SRW lead us to conclude that:
• the SRW cannot move from one level to another without passing through all values
between (“continuity”);
• all 2n length−n paths are equally likely so two events containing the same number
no. paths
, which allows the probability of one
of paths have the same probability,
2n
event to be determined by showing that the paths belonging to this event are in oneto-one correspondence with those of an event of known probability — in many cases
this correspondence is established geometrically, via reasoning know as reflection
principle.
•
Exercise 0.5 — Symmetric random walk
Prove that:
(a) P (X2 6= 0) = P (X2 = 0) = 12 ;
(b) P (Xn = k) = P (Xn = −k), for each n and k.
•
Proposition 0.6 — Expectation and symmetric random walk (Karr, 1993, p. 5)
The average value of any ith −step of a SRW is equal to
E(Yi ) = (−1) × P (Yi = −1) + (+1) × P (Yi = +1)
= 0.
(7)
5
Additivity of probability translates to linearity of expectation, thus the average position
equals
!
n
X
E(Xn ) = E
Yi
i=1
=
n
X
E(Yi )
i=1
= 0.
(8)
•
Proposition 0.7 — Conditioning and symmetric random walk (Karr, 1993, p. 6)
We can revise probability in light of the knowledge that some event has occurred. For
2n
example, we know that P (X2n = 0) = 2n
× 21 . However, if we knew that X2n−1 = 1
n
then the event {X2n = 0} occurs with probability 21 . In fact,
P (X2n−1 = 1, X2n = 0)
P (X2n−1 = 1)
P (X2n−1 = 1, Y2n = −1)
=
P (X2n−1 = 1)
P (X2n−1 = 1) × P (Y2n = −1)
=
P (X2n−1 = 1)
1
.
=
2
P (X2n = 0|X2n−1 = 1) =
Note that, since the steps Yi are independent random variables and X2n−1 =
we can state that Y2n is independent of X2n−1 .
(9)
P2n−1
i=1
Yi ,
•
Exercise 0.8 — Conditioning and asymmetric random walk5
Random walk models are often found in physics, from particle motion to a simple
description of a polymer.
A physicist assumes that the position of a particle at time n, Xn , is governed by
an asymmetric random walk — starting at 0 and with probability of an upward (resp.
downward) step equal to p (resp. 1 − p), where p ∈ (0, 1)\{ 12 }.
Derive P (X2n = 0|X2n−2 = 0), for n = 2, 3, . . .
•
5
Exam 2010/01/19.
6
Proposition 0.9 — Time of first return to the origin and symmetric random
walk (Karr, 1993, pp. 7-9)
The time at which the SRW first returns to the origin,
T 0 = min{n ∈ IN : Xn = 0},
(10)
is an important functional of the SRW (it maps the SRW into a scalar). It can represent
the time to ruin.
Interestingly enough, for n ∈ IN , T 0 must be even (recall that X0 = 0). And, for
n ∈ IN :
2n
2n
1
0
;
(11)
P (T > 2n) = P (X1 6= 0, . . . , X2n 6= 0) =
×
2
n
2n
2n
1
1
0
×
.
(12)
P (T = 2n) =
2n − 1 n
2
√
1
Moreover, using the Stirling’s approximation to n!, n! ' 2π nn+ 2 e−n , we get
P (T 0 < +∞) = 1.
(13)
P
1
If we note that P (T 0 > 2n) ' √1πn and recall that +∞
n=1 ns only converges for s ≥ 2,
we can conclude that T 0 assumes large values with probabilities large enough that
+∞
X
2n P (T 0 = 2n) = +∞ ⇒ E(T 0 ) = +∞.
(14)
n=1
•
Exercise 0.10 — Time of first return to the origin and symmetric random walk
(a) Prove result (12) using (11).
(b) Use the Stirling’s approximation to n!, n! '
√
1
2π nn+ 2 e−n to prove that
1
lim P (T 0 > 2n) = lim √ .
n→+∞
n→+∞
πn
(c) Use the previous result and the fact that
P (T 0 < +∞) = 1 − lim P (T 0 > 2n)
n→+∞
to derive (13).
P+∞
P
0
0
(d) Verify that
= 2n) = 1 + +∞
> 2n), even though we have
n=1 2n P (T
n=1 P (T
P+∞
E(Z) = 2 × 1 + n=1 P (Z > 2n) , for any positive and even random variable Z
P
with finite expected value E(Z) = +∞
•
n=1 2n × P (Z = 2n).
7
Proposition 0.11 — First passage times and symmetric random walk (Karr,
1993, pp. 9–11)
Similarly, the first passage time
T k = min{n ∈ IN : Xn = k},
(15)
has the following properties, for n ∈ IN , k ∈ {−n, . . . , −1, 1, . . . , n} and n mod 2 =
k mod 2:
|k|
× P (Xn = k);
n
P (T k < ∞) = 1;
P (T k = n) =
(16)
(17)
E(T k ) = +∞.
(18)
•
The following results pertain to the asymptotic behaviour of the position of a
symmetric random walk and to the fraction of time spent positive.
Proposition 0.12 — Law of large numbers (Karr, 1993, p. 12)
P
Let Yn and Xn = ni=1 Yi represent the size of the nth. step and the position at time n
of a random walk, respectively.
Xn
P
lim
= 0 = 1,
(19)
n→+∞ n
P
that is, the “empirical averages”, Xnn = n1 ni=1 Yi , converge to the “theoretical average”
E(Y1 ).
•
Proposition 0.13 — Central limit theorem (Karr, 1993, pp. 12–13)




 Xn − E Xn 
 Xn − E(Y )

1
n
n
n
q
q
lim P
lim P
≤x
≤ x = n→+∞
n→+∞



V (Y1 )
V Xnn
n
Z x
y2
1
√ e− 2 dy
=
2π
−∞
= Φ(x), x ∈ IR
(20)
So, for large values of n, difficult-to-compute probabilities can be approximated. For
instance, for a < b, we get:
8
P (a < Xn ≤ b) =
X
P (Xn = k)
a<k≤b


a −0
Xn
b
−0
− 0
n
n
n
q < q
= P
≤ q

1
1
1 
n
n
n
√
√
' Φ(b/ n) − Φ(a/ n).
(21)
•
Exercise 0.14 — Central limit theorem6
The words “symmetric random walk” refer to this situation.
The proverbial drunk (PD) is clinging to the lamppost. He decides to start
walking. The road runs east and west. In his inebriated state he is as likely
to take a step east (forward) as west (backward). In each new position he is
again as likely to go forward as backward. Each of his steps are of the same
length but of random direction — east or west.
http://www.physics.ucla.edu/∼chester/TECH/RandomWalk/3Pane.html
Admit that each step of PD has length equal to one meter and that he has already taken
exactly 100 (a hundred) steps.
Find an approximate value for the probability that PD is within a five meters
neighborhood of the lamppost.
•
Proposition 0.15 — Arc sine law (Karr, 1993, pp. 13–14)
P
The fraction of time spent positive Wnn = n1 ni=1 IIN (Xi + Xi−1 ) has the following limiting
law:7
√
Wn
2
lim P
≤ x = arcsin x.
(22)
n→+∞
n
π
Moreover, the associated limiting density function, √ 1
, is a U −shaped density.
π
x(1−x)
Thus, Wnn is more likely to be near 0 or 1 than near 1/2.
Please note that we can get the limiting distribution function by using the Stirling’s
approximation and the following result:
2n
2k
2n − 2k
1
.
(23)
P (W2n = 2k) =
×
×
2
k
n−k
•
6
7
Exam 2010/02/04.
Please note that we need Xi−1 to be able to count the last step just before hitting zero from above.
9
Exercise 0.16 — Arc sine law
Prove result (22).
•
Exercise 0.17 — Arc sine law8
The random walk hypothesis is due to French economist Louis Bachelier (1870–1946) and
asserts that the random nature of a commodity or stock prices cannot reveal trends and
therefore current prices are no guide to future prices. Surprisingly, an investor assumes
that his/her daily financial score is governed by a symmetric random walk starting at 0.
Obtain the corresponding approximate value for the probability that the fraction of
time the financial score is positive exceeds 50%.
•
Exercise 0.18 — The cliff-hanger problem (Mosteller, 1965, pp. 51–54)
From where he stands (X0 = 1), one step toward the cliff would send the drunken man
over the edge. He takes random steps, either toward or away from the cliff. At any step,
his probability of taking a step away is p and of a step toward the cliff 1 − p.
What is his chance of not escaping the cliff? (Write the results in terms of p.)
•
References
• Grimmett, G.R. and Stirzaker, D.R. (2001). Probability and Random Processes
(3rd. edition). Oxford. (QA274.12-.76.GRI.40695 refers to the library code of the
1st. and 2nd. editions from 1982 and 1992, respectively.)
• Karr, A.F. (1993). Probability. Springer-Verlag.
8
Test 2009/11/07.
10
Chapter 1
Probability spaces
[...] have been taught that the universe evolves according to deterministic
laws that specify exactly its future, and a probabilistic description is necessary
only because of our ignorance. This deep-rooted skepticism in the validity
of probabilistic results can be overcome only by proper interpretation of the
meaning of probability.
Papoulis (1965, p. 3)
Probability is the mathematics of uncertainty. It has flourished
under the stimulus of applications, such as insurance, demography, [...], clinical
trials, signal processing, [...], spread of infectious diseases, [...], medical
imaging, etc. and have furnished both mathematical questions and genuine
interest in the answers.
Karr (1993, p. 15)
Much of our life is based on the belief that the future is largely unpredictable
(Grimmett and Stirzaker, 2001, p. 1), nature is liable to change and chance governs
life.
We express this belief in chance behaviour by the use of words such as random, probable
(probably), probability, likelihood (likeliness), etc.
There are essentially four ways of defining probability (Papoulis, 1965, p. 7) and this
is quite a controversial subject, proving that not all of probability and statistics is cutand-dried (Righter, 200–):
• a priori definition as a ratio of favorable to total number of alternatives (classical
definition; Laplace);1
1
See the first principle of probability in http://en.wikipedia.org/wiki/Pierre-Simon Laplace
11
• relative frequency (Von Mises);2
• probability as a measure of belief (inductive reasoning,3 subjective probability;
Bayesianism);4
• axiomatic (measure theory; Kolmogorov’s axioms).5
Classical definition of probability
The classical definition of probability of an event A is found a priori without actual
experimentation, by counting the total number N = #Ω < +∞ of possible outcomes
of the random experiment. If these outcomes are equally likely and NA = #A of these
outcomes the event A occurs, then
P (A) =
#A
NA
=
.
N
#Ω
(1.1)
Criticism of the classical definition of probability
It is only holds if N = #Ω < +∞ and all the N outcomes are equally likely. Moreover,
• serious problems often arise in determining N = #Ω;
• it can be used only for a limited class of problems since the equally likely condition
is often violated in practice;
• the classical definition, although presented as a priori logical necessity, makes
implicit use of the relative-frequency interpretation of probability;
• in many problems the possible number of outcomes is infinite, so that to determine
probabilities of various events one must introduce some measure of length or area.
2
Kolmogorov said: “[...] mathematical theory of probability to real ’random phenomena’ must depend
on some form of the frequency concept of probability, [...] which has been established by von Mises [...].”
(http://en.wikipedia.org/wiki/Richard von Mises)
3
Inductive reasoning or inductive logic is a type of reasoning which involves moving from a set of
specific facts to a general conclusion (http://en.wikipedia.org/wiki/Inductive reasoning).
4
Bayesianism uses probability theory as the framework for induction. Given new evidence, Bayes’
theorem is used to evaluate how much the strength of a belief in a hypothesis should change with the
data we collected.
5
http://en.wikipedia.org/wiki/Kolmogorov axioms
12
Relative frequency definition of probability
The relative frequency approach was developed by Von Mises in the beginning of the 20th.
century; at that time the prevailing definition of probability was the classical one and his
work was a healthy alternative (Papoulis, 1965, p. 9).
The relative frequency definition of probability used to be popular among engineers
and physicists. A random experiment is repeated over and over again, N times; if the
event A occurs NA times out of N , then the probability of A is defined as the limit of the
relative frequency of the occurrence of A:
NA
.
N →+∞ N
(1.2)
P (A) = lim
Criticism of the relative frequency definition of probability
This notion is meaningless in most important applications, e.g. finding the probability of
the space shuttle blowing up, or of an earthquake (Righter, 200–), essentially because we
cannot repeat the experiment.
It is also useless when dealing with hypothetical experiments (e.g. visiting Jupiter).
Subjective probability, personal probability, Bayesian approach; criticism
Each person determines for herself what the probability of an event is; this value is in
[0, 1] and expresses the personal belief on the occurrence of the event.
The Bayesian approach is the approach used by most engineers and many scientists and
business people. It bothers some, because it is not “objective”. For a Bayesian, anything
that is unknown is random, and therefore has a probability, even events that have already
occurred. (Someone flipped a fair coin in another room, the chance that it was heads or
tails is .5 for a Bayesian. A non-Bayesian could not give a probability.)
With a Bayesian approach it is possible to include nonstatistical information (such as
expert opinions) to come up with a probability. The general Bayesian approach is to come
up with a prior probability, collect data, and use the data to update the probability (using
Bayes’ Law, which we will study later).
(Righter, 200–)
To understand the (axiomatic) definition of probability we shall need the following
concepts:
• random experiment, whose outcome cannot be determined in advance;
• sample space Ω, the set of all (conceptually) possible outcomes;
13
• outcomes ω, elements of the sample space, also referred to as sample points or
realizations;
• events A, a set of outcomes;
• σ−algebra on Ω, a family of subsets of Ω containing Ω and closed under
complementation and countable union.
14
1.1
Random experiments
Definition 1.1 — Random experiment
A random experiment consists of both a procedure and observations,6 and its outcome
cannot be determined in advance.
•
There is some uncertainty in what will observed in the random experiment, otherwise
performing the experiment would be unnecessary.
Example 1.2 — Random experiments
Random experiment
E1
Give a lecture.
Observe the number of students seated in the 4th. row, which has 7 seats.
E2
Choose a highway junction.
Observe the number of car accidents in 12 hours.
E3
Walk to a bus stop.
Observe the time (in minutes) you wait for the arrival of a bus.
E4
Give n lectures.
Observe the number of students seated in the forth row in each of those n lectures.
E5
Consider a particle in a gas modeled by a random walk.
Observe the steps at times 1, 2, . . .
E6
Consider a cremation chamber.
Observe the temperature in the center of the chamber over the interval of time [0, 1].
•
Exercise 1.3 — Random experiment
Identify at least one random experiment based on your daily schedule.
•
Definition 1.4 — Sample space (Yates and Goodman, 1999, p. 8)
The sample space Ω of a random experiment is the finest-grain, mutually exclusive,
collectively exhaustive set of all possible outcomes of the random experiment.
•
6
Yates and Goodman (1999, p. 7).
15
The finest-grain property simply means that all possible distinguishable outcomes are
identified separately. Moreover, Ω is (usually) known before the random experiment takes
place. The choice of Ω balances fidelity to reality with mathematical convenience (Karr,
1993, p. 12).
Remark 1.5 — Categories of sample spaces (Karr, 1993, p. 16–17)
In practice, most sample spaces fall into one of the six categories:
• Finite set
The simplest random experiment has two outcomes.
A random experiment with n possible outcomes may be modeled with a sample
space consisting of n integers.
• Countable set
The sample space for an experiment with countably many possible outcomes is
ordinarily the set IN = {1, 2, . . .} of positive integers or the set of {. . . , −1, 0, +1, . . .}
of all integers.
Whether a finite or a countable sample space better describes a given phenomenon
is a matter of judgement and compromise. (Comment!)
• The real line IR (and intervals in IR)
The most common sample space is the real line IR (or the unit interval [0, 1] the
nonnegative half-line IR0+ ), which is used for most all numerical phenomena that are
not inherently integer-valued.
• Finitely many replications
Some random experiments result from the n (n ∈ IN ) replications of a basic
experiment with sample space Ω0 . In this case the sample space is the Cartesian
product Ω = Ωn0 .
• Infinitely many replications
If a basic random experiment is repeated infinitely many times we deal with the
sample space Ω = ΩI0N .
• Function spaces
In some random experiments the outcome is a trajectory followed by a system over
an interval of time. In this case the outcomes are functions.
•
16
Example 1.6 — Sample spaces
The sample spaces defined below refer to the random experiments defined in Example
1.2:
Random experiment
Sample space (Ω)
Classification of Ω
E1
{0, 1, 2, 3, 4, 5, 6, 7}
Finite set
E2
IN0 = {0, 1, 2, . . .}
Countable set
E3
IR0+
Interval in IR
E4
{0, 1, 2, 3, 4, 5, 6, 7}n
Finitely many replications
E5
{−1, +1}IN
Infinitely many replications
E6
C([0, 1])
Function space
Note that C([0, 1]) represents the vector space of continuous, real-valued functions on
[0, 1].
•
17
1.2
Events and classes of sets
Definition 1.7 — Event (Karr, 1993, p. 18)
Given a random experiment with sample space Ω, an event can be provisionally defined
as a subset of Ω whose probability is defined.
•
Remark 1.8 — An event A occurs if the outcome ω of the random experiment belongs
to A, i.e. ω ∈ A.
•
Example 1.9 — Events
Some events associated to the six random experiments described in examples 1.2 and 1.6:
E.A.
E1
Event
A = “observe at least 3 students in the 4th. row”
= {3, . . . , 7}
E2
B = “observe more than 4 car accidents in 12 hours”
= {5, 6, . . .}
E3
C = “wait more than 8 minutes”
= (8, +∞)
E4
D = “observe at least 3 students in the 4th. row, in 5 consecutive days”
= {3, . . . , 7}5
E5
E = “an ascending path”
= {(1, 1, . . .)}
E6
F = “temperature above 250o over the interval [0, 1]”
= {f ∈ C([0, 1]) : f (x) > 250, x ∈ [0, 1]}
•
Definition 1.10 — Set operations (Resnick, 1999, p. 3)
As subsets of the sample space Ω, events can be manipulated using set operations. The
set operations which you should know and will be commonly used are listed next:
• Complementation
The complement of an event A ⊂ Ω is
Ac = {ω ∈ Ω : ω 6∈ A}.
(1.3)
18
• Intersection
The intersection of events A and B (A, B ⊂ Ω) is
A ∩ B = {ω ∈ Ω : ω ∈ A and ω ∈ B}.
(1.4)
The events A and B are disjoint (mutually exclusive) if A ∩ B = ∅, i.e. they have
no outcomes in common, therefore they never happen at the same time.
• Union
The union of events A and B (A, B ⊂ Ω) is
A ∪ B = {ω ∈ Ω : ω ∈ A or ω ∈ B}.
(1.5)
Karr (1993) uses A + B to denote A ∪ B when A and B are disjoint.
• Set difference
Given two events A and B (A, B ⊂ Ω), the set difference between B and A consists
of those outcomes in B but not in A:
B\A = B ∩ Ac .
(1.6)
• Symmetric difference
Let A and B be two events (A, B ⊂ Ω). Then the outcomes that are in one but not
in both sets consist on the symmetric difference:
A∆B = (A\B) ∪ (B\A).
(1.7)
•
Exercise 1.11 — Set operations
Represent the five set operations in Definition 1.10 pictorially by Venn diagrams.
•
Proposition 1.12 — Properties of set operations (Resnick, 1999, pp. 4–5)
Set operations satisfy well known properties such as commutativity, associativity, De
Morgan’s laws, etc., providing now and then connections between set operations. These
properties have been condensed in the following table:
19
Set operation
Property
Complementation
(Ac )c = A
∅c = Ω
Ωc = ∅
Intersection and union
Commutativity
A ∩ B = B ∩ A, A ∪ B = B ∪ A
A ∩ ∅ = ∅, A ∪ ∅ = A
A ∩ A = A, A ∪ A = A
A ∩ Ω = A, A ∪ Ω = Ω
A ∩ Ac = ∅, A ∪ Ac = Ω
Associativity
(A ∩ B) ∩ C = A ∩ (B ∩ C)
(A ∪ B) ∪ C = A ∪ (B ∪ C)
De Morgan’s laws
(A ∩ B)c = Ac ∪ B c
(A ∪ B)c = Ac ∩ B c
Distributivity
(A ∩ B) ∪ C = (A ∪ C) ∩ (B ∪ C)
(A ∪ B) ∩ C = (A ∩ C) ∪ (B ∩ C)
•
Definition 1.13 — Relations between sets (Resnick, 1999, p. 4)
Now we list ways sets A and B can be compared:
• Set containment or inclusion
A is a subset of B, written A ⊂ B or B ⊃ A, iff A ∩ B = A. This means that
ω ∈ A ⇒ ω ∈ B.
(1.8)
So if A occurs then B also occurs. However, the occurrence of B does not imply
the occurrence of A.
• Equality
Two events A and B are equal, written A = B, iff A ⊂ B and B ⊂ A. This means
ω ∈ A ⇔ ω ∈ B.
(1.9)
•
20
Proposition 1.14 — Properties of set containment (Resnick, 1999, p. 4)
These properties are straightforward but we stated them for the sake of completeness and
their utility in the comparison of the probabilities of events:
• A⊂A
• A ⊂ B, B ⊂ C ⇒ A ⊂ C
• A ⊂ C, B ⊂ C ⇒ (A ∪ B) ⊂ C
• A ⊃ C, B ⊃ C ⇒ (A ∩ B) ⊃ C
• A ⊂ B ⇔ B c ⊂ Ac .
•
These properties will be essential to calculate or relate probabilities of (sophisticated)
events.
Remark 1.15 — The jargon of set theory and probability theory
What follows results from minor changes of Table 1.1 from Grimmett and Stirkazer (2001,
p. 3):
Typical notation
Set jargon
Probability jargon
Ω
Collection of objects
Sample space
ω
Member of Ω
Outcome
A
Subset of Ω
Event (that some outcome in A occurs)
Ac
Complement of A
Event (that no outcome in A occurs)
A∩B
Intersection
A and B occur
A∪B
Union
Either A or B or both A and B occur
B\A
Difference
B occurs but not A
A∆B
Symmetric difference
Either A or B, but not both, occur
A⊂B
Inclusion
If A occurs then B occurs too
∅
Empty set
Impossible event
Ω
Whole space
Certain event
•
21
Functions on the sample space (such as random variables defined in the next chapter)
are even more important than events themselves.
An indicator function is the simplest way to associate a set with a (binary) function.
Definition 1.16 — Indicator function (Karr, 1993, p. 19)
The indicator function of the event A ⊂ Ω is the function on Ω given by
(
1 if w ∈ A
1A (w) =
0 if w 6∈ A
(1.10)
•
Therefore, 1A indicates whether A occurs.
The indicator function of an event, which resulted from a set operation on events A
and B, can often be written in terms of the indicator functions of these two events.
Proposition 1.17 — Properties of indicator functions (Karr, 1993, p. 19)
Simple algebraic operations on the indicator functions of the events A and B translate
set operations on these two events:
1A c = 1 − 1A
(1.11)
1A∩B = min{1A , 1B }
= 1A × 1B
(1.12)
1A∪B = max{1A , 1B };
(1.13)
1B\A = 1B∩Ac
= 1B × (1 − 1A )
(1.14)
1A∆B = |1A − 1B |.
(1.15)
•
Exercise 1.18 — Indicator functions
Solve exercises 1.1 and 1.7 of Karr (1993, p. 40).
•
The definition of indicator function quickly yields the following result when we are
able compare events A and B.
22
Proposition 1.19 — Another property of indicator functions (Resnick, 1999, p.
5)
Let A and B be two events of Ω. Then
A ⊆ B ⇔ 1A ≤ 1B .
(1.16)
Note here that we use the convention that for two functions f , g with domain Ω and range
IR, we have f ≤ g iff f (ω) ≤ g(ω) for all ω ∈ Ω.
•
Motivation 1.20 — Limits of sets (Resnick, 1999, p. 6)
The definition of convergence concepts for random variables rests on manipulations of
sequences of events which require the definition of limits of sets.
•
Definition 1.21 — Operations on sequences of sets (Karr, 1993, p. 20)
Let (An )n∈IN be a sequence of events of Ω. Then the union and the intersection of (An )n∈IN
are defined as follows
+∞
[
n=1
+∞
\
An = {ω : ω ∈ An for some n}
(1.17)
An = {ω : ω ∈ An for all n}.
(1.18)
n=1
The sequence (An )n∈IN is said to be pairwise disjoint if Ai ∩ Aj = ∅ whenever i 6= j.
•
Definition 1.22 — Lim sup, lim inf and limit set (Karr, 1993, p. 20)
Let (An )n∈IN be a sequence of events of Ω. Then we define the two following limit sets:
lim sup An =
+∞
\ +∞
[
An
(1.19)
k=1 n=k
= {ω ∈ Ω : ω ∈ An for infinitely many values of n}
= {An , i.o.}
lim inf An =
+∞
[ +∞
\
An
(1.20)
k=1 n=k
= {ω ∈ Ω : ω ∈ An for all but finitely many values of n}
= {An , ult.},
where i.o. and ult. stand for infinitely often and ultimately, respectively.
23
Let A be an event of Ω. Then the sequence (An )n∈IN is said to converge to A, written
An → A or limn→+∞ An = A, if
lim inf An = lim sup An = A.
(1.21)
•
Example 1.23 — Lim sup, lim inf and limit set
Let (An )n∈IN be a sequence of events of Ω such that
(
A for n even
An =
Ac for n odd.
(1.22)
Then
lim sup An =
+∞
\ +∞
[
An
k=1 n=k
= Ω
(1.23)
6=
lim inf An =
+∞
[ +∞
\
An
k=1 n=k
= ∅,
(1.24)
so there is no limit set limn→ An .
•
Exercise 1.24 — Lim sup, lim inf and limit set
Solve Exercise 1.3 of Karr (1993, p. 40).
•
Proposition 1.25 — Properties of lim sup and lim inf (Resnick, 1999, pp. 7–8)
Let (An )n∈IN be a sequence of events of Ω. Then
lim inf An ⊂ lim sup An
(1.25)
(lim inf An )c = lim sup(Acn ).
(1.26)
•
24
Definition 1.26 — Monotone sequences of events (Resnick, 1999, p. 8)
Let (An )n∈IN be a sequence of events of Ω. It is said to be monotone non-decreasing,
written An ↑, if
A1 ⊆ A2 ⊆ A3 ⊆ . . . .
(1.27)
(An )n∈IN is monotone non-increasing, written An ↓, if
A1 ⊇ A2 ⊇ A3 ⊇ . . . .
(1.28)
•
Proposition 1.27 — Properties of monotone sequences of events (Karr, 1993,
pp. 20–21)
Suppose (An )n∈IN be a monotone sequence of events. Then
An ↑ ⇒
An ↓ ⇒
lim An =
n→+∞
lim An =
n→+∞
+∞
[
n=1
+∞
\
An
(1.29)
An .
(1.30)
n=1
•
Exercise 1.28 — Properties of monotone sequences of events
Prove Proposition 1.27.
•
Example 1.29 — Monotone sequences of events
The Galton-Watson process is a branching stochastic process arising from Francis Galton’s
statistical investigation of the extinction of family names. Modern applications include the
survival probabilities for a new mutant gene, [...], or the dynamics of disease outbreaks
in their first generations of spread, or the chances of extinction of small populations of
organisms. (http://en.wikipedia.org/wiki/Galton-Watson process)
Let (Xn )IN0 be a stochastic process, where Xn represents the size of generation n. A
(Xn )IN0 is Galton-Watson process if it evolves according to the recurrence formula:
• X0 = 1 (we start with one individual); and
P n (n)
(n)
• Xn+1 = X
the number of descendants of
i=1 Zi , where, for each n, Z
i represents
(n)
the individual i from generation n and Zi
is a sequence of i.i.d. non-negative
i∈IN
random variables.
25
Let An = {Xn = 0}. Since A1 ⇒ A2 ⇒ . . ., i.e. (An )n∈IN is a non-decreasing monotone
S
sequence of events, written An ↑, we get An → A = +∞
n=1 An . Moreover, the extinction
probability is given by
!
+∞
[
P ({Xn = 0 for some n}) = P
{Xn = 0} = P
lim {Xn = 0}
n=1
+∞
[
= P
n→+∞
!
An
n=1
= P
lim An .
n→+∞
(1.31)
Later on, we shall conclude that we can conveniently interchange the limit sign and
the probability function and add: P (Xn = 0 for some n) = P (limn→+∞ {Xn = 0}) =
limn→+∞ P ({Xn = 0}).
•
Proposition 1.30 — Limits of indicator functions (Karr, 1993, p. 21)
In terms of indicator functions,
An → A ⇔ 1An (w) → 1A (w), ∀w ∈ Ω.
(1.32)
Thus, the convergence of sets is the same as pointwise convergence of their indicator
functions.
•
Exercise 1.31 — Limits of indicator functions (Exercise 1.8, Karr, 1993, p. 40)
Prove Proposition 1.30.
•
Motivation 1.32 — Closure under set operations (Resnick, 1999, p. 12)
We need the notion of closure because we want to combine and manipulate events to make
more complex events via set operations and we require that certain set operations do not
carry events outside the family of events.
•
Definition 1.33 — Closure under set operations (Resnick, 1999, p. 12)
Let C be a collection of subsets of Ω. C is closed under a set operation7 if the set obtained
by performing the set operation on sets in C yields a set in C.
•
7
Be it a countable union, finite union, countable intersection, finite intersection, complementation,
monotone limits, etc.
26
Example 1.34 — Closure under set operations (Resnick, 1999, p. 12)
• C is closed under finite union if for any finite collection A1 , . . . , An of sets in C,
Sn
i=1 Ai ∈ C.
• Suppose Ω = IR and C = {finite real intervals} = {(a, b] : −∞ < a < b < +∞}.
Then C is not closed under finite unions since (1, 2] ∪ (36, 37] is not a finite interval.
However, C is closed under intersection since (a, b]∩(c, d] = (max{a, c}, min{b, d}] =
(a ∨ c, b ∧ d].
• Consider now Ω = IR and C = {open real subsets}. C is not closed under
complementation since the complement of as open set is not open.
•
Definition 1.35 — Algebra (Resnick, 1999, p. 12)
A is an algebra (or field) on Ω if it is a non-empty class of subsets of Ω closed under finite
union, finite intersection and complementation.
A minimal set of postulates for A to be an algebra on Ω is:
1. Ω ∈ A
2. A ∈ A ⇒ Ac ∈ A
3. A, B ∈ A ⇒ A ∪ B ∈ A.
•
Remark 1.36 — Algebra
Please note that, by the De Morgan’s laws, A is closed under finite intersection ((A∪B)c =
Ac ∩ B c ∈ A), thus we do not need a postulate concerning finite intersection.
•
Motivation 1.37 — σ-algebra (Karr, 1993, p. 21)
To define a probability function dealing with an algebra is not enough: we need to define
a collection of sets which is closed under countable union, countable intersection, and
complementation.
•
Definition 1.38 — σ−algebra (Resnick, 1999, p. 12)
F is a σ−algebra on Ω if it is a non-empty class of subsets of Ω closed under countable
union, countable intersection and complementation.
A minimal set of postulates for F to be an σ−algebra on Ω is:
27
1. Ω ∈ F
2. A ∈ F ⇒ Ac ∈ F
3. A1 , A2 , . . . ∈ F ⇒
S+∞
i=1
Ai ∈ F.
•
Example 1.39 — σ−algebra (Karr, 1993, p. 21)
• Trivial σ−algebra
F = {∅, Ω}
• Power set
F = IP (Ω) = class of all subsets of Ω
In general, neither of these two σ−algebras is specially interesting or useful — we need
something in between.
•
Definition 1.40 — Generated σ−algebra (http://en.wikipedia.org/wiki/Sigmaalgebra)
If U is an arbitrary family of subsets of Ω then we can form a special σ−algebra
containing U, called the σ−algebra generated by U and denoted by σ(U), by intersecting
all σ−algebras containing U.
Defined in this way σ(U) is the smallest/minimal σ−algebra on Ω that contains U. •
Example 1.41 — Generated
algebra; Karr, 1993, p. 22)
σ−algebra
(http://en.wikipedia.org/wiki/Sigma-
• Trivial example
Let Ω = {1, 2, 3} and U = {{1}}. Then σ(U) = {∅, {1}, {2, 3}, Ω} is a σ−algebra
on Ω.
• σ−algebra generated by a finite partition
If U = {A1 , . . . , An } is a finite partition of Ω — that is, A1 , . . . , An are disjoint and
Sn
S
i=1 Ai = Ω — then σ(U) = { i∈I Ai : I ⊆ {1, . . . , n}} which includes ∅.
• σ−algebra generated by a countable partition
If U = {A1 , A2 , . . .} is a countable partion of Ω — that is, A1 , A2 , , . . . are disjoint
S
S
and +∞
•
i=1 Ai = Ω — then σ(U) = { i∈I Ai : I ⊆ IN } which also includes ∅.
28
Since we tend to deal with real random variables we have to define a σ−algebra on
Ω = IR and the power set on IR, IP (IR) is not an option. The most important σ−algebra
on IR is the one defined as follows.
Definition 1.42 — Borel σ−algebra on IR (Karr, 1993, p. 22)
The Borel σ−algebra on IR, denoted by B(IR), is generated by the class of intervals
U = {(a, b] : −∞ < a < b < +∞},
(1.33)
that is, σ(U) = B(IR). Its elements are called Borel sets.8
•
Remark 1.43 — Borel σ−algebra on IR (Karr, 1993, p. 22)
• Every “reasonable” set of IR — such as intervals, closed sets, open sets, finite sets,
T
and countable sets — belong to B(IR). For instance, {x} = +∞
n=1 (x − 1/n, x].
• Moreover, the Borel σ−algebra on IR, B(IR), can also be generated by the class of
intervals {(−∞, a] : −∞ < a < +∞} or {(b, +∞) : −∞ < b < +∞}.
• B(IR) 6= IP (IR).
•
Definition 1.44 — Borel σ−algebra on IRd (Karr, 1993, p. 22)
The Borel σ−algebra on IRd , d ∈ IN , B(IRd ), is generated by the class of rectangles that
are Cartesian products of real intervals
( d
)
Y
U=
(ai , bi ] : −∞ < ai < bi < +∞, i = 1, . . . , d .
(1.34)
i=1
•
Exercise 1.45 — Generated σ−algebra (Exercise 1.9, Karr, 1993, p. 40)
Given sets A and B of Ω, identify all sets in σ({A, B}).
•
Exercise 1.46 — Borel σ−algebra on IR (Exercise 1.10, Karr, 1993, p. 40)
Prove that {x} is a Borel set for every x ∈ IR.
•
8
Borel sets are named after Émile Borel.
Along with René-Louis Baire and Henri
Lebesgue, he was among the pioneers of measure theory and its application to probability theory
(http://en.wikipedia.org/wiki/Émile Borel).
29
1.3
Probabilities and probability functions
Motivation 1.47 — Probability function (Karr, 1993, p. 23)
A probability is a set function, defined for events; it should be countably additive (i.e.
σ−additive), that is, the probability of a countable union of disjoint events is the sum of
their individual probabilities.
•
Definition 1.48 — Probability function (Karr, 1993, p. 24)
Let Ω be the sample space and F be the σ−algebra of events of Ω. A probability on
(Ω, F) is a function P : Ω → IR such that:
1. Axiom 1
9
— P (A) ≥ 0, ∀A ∈ F.
2. Axiom 2 — P (Ω) = 1.
3. Axiom 3 (countable additivity or σ−additivity)
Whenever A1 , A2 , . . . are (pairwise) disjoint events in F,
! +∞
+∞
[
X
P
Ai =
P (Ai ).
i=1
(1.35)
i=1
•
Remark 1.49 — Probability function
The probability function P transforms events in real numbers in [0, 1].
•
Definition 1.50 — Probability space (Karr, 1993, p. 24)
The triple (Ω, F, P ) is a probability space.
•
Example 1.51 — Probability function (Karr, 1993, p. 25)
Let
• {A1 , . . . , An } be a finite partition of Ω — that is, A1 , . . . , An are (nonempty and
S
pairwise) disjoint events and ni=1 Ai = Ω;
• F be the σ−algebra generated by the finite partition {A1 , A2 , . . . , An }, i.e. F =
σ({A1 , . . . , An });
P
• p1 , . . . , pn positive numbers such that ni=1 pi = 1.
9
Righter (200—) called the first and second axioms duh rules.
30
Then the function defined as
!
[
X
P
Ai =
pi , ∀I ⊆ {1, . . . , n},
i∈I
(1.36)
i∈I
where pi = P (Ai ), is a probability function on (Ω, F).
•
Exercise 1.52 — Probability function (Exercise 1.11, Karr, 1993, p. 40)
Let A, B and C be disjoint events such that: A ∪ B ∪ C = Ω; P (A) = 0.6, P (B) = 0.3
and P (C) = 0.1. Calculate all probabilities of all events in σ({A, B, C}).
•
Motivation 1.53 — Elementary properties of probability functions
The axioms do not teach us how to calculate the probabilities of events. However, they
establish rules for their calculation such as the following ones.
•
Proposition 1.54 — Elementary properties of probability functions (Karr, 1993,
p. 25)
Let (Ω, F, P ) be a probability space then:
1. Probability of the empty set
P (∅) = 0.
(1.37)
2. Finite additivity
If A1 , . . . , An are (pairwise) disjoint events then
!
n
n
X
[
P
P (Ai ).
Ai =
i=1
(1.38)
i=1
Probability of the complement of A
Consequently, for each A,
P (Ac ) = 1 − P (A).
(1.39)
3. Monotonicity of the probability function
If A ⊆ B then
P (B \ A) = P (B) − P (A).
(1.40)
Therefore if A ⊆ B then
P (A) ≤ P (B).
(1.41)
31
4. Addition rule
For all A and B (disjoint or not),
P (A ∪ B) = P (A) + P (B) − P (A ∩ B).
(1.42)
•
Remark 1.55 — Elementary properties of probability functions
According to Righter (200—), (1.41) is another duh rule but adds one of Kahneman and
Tversky’s most famous examples, the Linda problem.
Subjects were told the story (in 70’s):
• Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy.
As a student she was deeply concerned with issues of discrimination and social justice
and also participated in anti nuclear demonstrations.
They are asked to rank the following statements by their probabilities:
• Linda is a bank teller.
• Linda is a bank teller who is active in the feminist movement.
Kahneman and Tversky found that about 85% of the subjects ranked “Linda is a bank
teller and is active in the feminist movement” as more probable than “Linda is a bank
teller”.
•
Exercise 1.56 — Elementary properties of probability functions
Prove properties 1. through 4. of Proposition 1.54 and that
P (B \ A) = P (B) − P (A ∩ B)
(1.43)
P (A∆B) = P (A ∪ B) − P (A ∩ B) = P (A) + P (B) − 2 × P (A ∩ B).
(1.44)
Hints (Karr, 1993, p. 25):
• property 1. can be also proved by using the finite additivity;
• property 2. by considering An+1 = An+2 = . . . = ∅;
• property 3. by rewriting B as (B \ A) ∪ (A ∩ B) = (B \ A) ∪ A;
• property 4. by rewriting A ∪ B as (A \ B) ∪ (A ∩ B) ∪ (B \ A).
32
•
We proceed with some advanced properties of probability functions.
Proposition 1.57 — Boole’s inequality or σ−subadditivity (Karr, 1993, p. 26)
Let A1 , A2 , . . . be events in F. Then
! +∞
+∞
[
X
P
An ≤
P (An ).
(1.45)
n=1
n=1
•
Exercise 1.58 — Boole’s inequality or σ−subadditivity
Prove Boole’s inequality by using the disjointification technique (Karr, 1993, p. 26),10 the
S
fact that Bn = An \ n−1
i=1 Ai ⊆ An , and by applying the σ−additivity and monotonicity
of probability functions.
•
Proposition 1.59 — Finite subadditivity (Resnick, 1999, p. 31)
The probability function P is finite subadditive in the sense that
!
n
n
X
[
P
P (Ai ),
Ai ≤
(1.46)
i=1
i=1
•
for all events A1 , . . . , An .
Remark 1.60 — Finite additivity
Finite additivity is a consequence of Boole’s inequality (i.e. σ−subadditivity). However,
finite additivity does not imply σ−subadditivity.
•
Proposition 1.61 — Inclusion-exclusion formula (Resnick, 1999, p. 30)
If A1 , . . . , An are events, then the probability of their union can be written as follows:
!
n
n
[
X
X
X
P
Ai
=
P (Ai ) −
P (Ai ∩ Aj ) +
P (Ai ∩ Aj ∩ Ak )
i=1
i=1
1≤i<j≤n
n
1≤i<j<k≤n
− . . . − (−1) × P (A1 ∩ . . . ∩ An ).
(1.47)
•
10
Note that
S+∞
n=1
An =
S+∞
n=1
Bn , where Bn = An \
33
S
n−1
i=1
Ai are disjoint events.
Remark 1.62 — Inclusion-exclusion formula (Resnick, 1999, p. 30)
The terms on the right side of (1.47) alternate in sign and give inequalities called
Bonferroni inequalities when we neglect the remainders. Two examples:
!
n
n
[
X
P
Ai
≤
P (Ai )
(1.48)
P
i=1
n
[
!
Ai
≥
i=1
i=1
n
X
P (Ai ) −
i=1
X
P (Ai ∩ Aj ).
(1.49)
1≤i<j≤n
•
Exercise 1.63 — Inclusion-exclusion formula
Prove the inclusion-exclusion formula by induction using the addition rule for n = 2
(Resnick, 1999, p. 30).
•
Proposition 1.64 — Monotone continuity (Resnick, 1999, p. 31)
Probability functions are continuous for monotone sequences of events in the sense that:
1. If An ↑ A, where An ∈ F, then P (An ) ↑ P (A).
2. If An ↓ A, where An ∈ F, then P (An ) ↓ P (A).
•
Exercise 1.65 — Monotone continuity
Prove Proposition 1.64 by using the disjointification technique, the monotone character
of the sequence of events and σ−additivity (Resnick, 1999, p. 31).
For instance property 1. can proved as follows:
• A1 ⊂ A2 ⊂ A3 ⊂ . . . ⊂ An ⊂ . . ..
• B1 = A1 , B2 = A2 \ A1 , B3 = A3 \ (A1 ∪ A2 ), . . . , Bn = An \
events.
Sn−1
i=1
Ai are disjoint
S
• Since A1 , A2 , . . . is a non-decreasing sequence of events An ↑ A = +∞
n=1 An =
S+∞
Sn
n=1 Bn , Bn = An \ An−1 , and
i=1 Bi = An . If we add to this σ−additivity,
we conclude that
!
! +∞
+∞
+∞
[
[
X
P (A) = P
An = P
Bn =
P (Bn )
n=1
=
lim ↑
n→+∞
n=1
n
X
i=1
P (Bi ) = lim ↑ P
n→+∞
n=1
n
[
i=1
!
Bi
= lim ↑ P (An ).
n→+∞
•
34
Motivation 1.66 — σ−additivity as a result of finite additivity and monotone
continuity (Karr, 1993, p. 26)
The next theorem shows that σ − additivity is equivalent to the confluence of finite
additivity (which is reasonable) and monotone continuity (which is convenient and
desirable mathematically).
•
Theorem 1.67 — σ−additivity as a result of finite additivity and monotone
continuity (Karr, 1993, p. 26)
Let P be a nonnegative, finitely additive set function on F with P (Ω) = 1. Then, the
following are equivalent:
1. P is σ−additive (thus a probability function).
2. Whenever An ↑ A in F, P (An ) ↑ P (A).
3. Whenever An ↓ A in F, P (An ) ↓ P (A).
4. Whenever An ↓ ∅ in F, P (An ) ↓ 0.
•
Exercise 1.68 — σ−additivity as a result of finite additivity and monotone
continuity
Prove Theorem 1.67.
Note that we need to prove 1. ⇒ 2. ⇒ 3. ⇒ 4. ⇒ 1. But since 2. ⇔ 3. by
complementation and 4. is a special case of 3. we just need to prove that 1. ⇒ 2. and
4. ⇒ 1. (Karr, 1993, pp. 26-27).
•
Remark 1.69 — Inf, sup, lim inf and lim sup
Let a1 , a2 , . . . be a sequence of real numbers. Then
• Infimum
The infimum of the set {a1 , a2 , . . .} — written inf an — corresponds to the greatest
element (not necessarily in {a1 , a2 , . . .}) that is less than or equal to all elements of
{a1 , a2 , . . .}.11
• Supremum
The supremum of the set {a1 , a2 , . . .} — written sup an — corresponds to the
smallest element (not necessarily in {a1 , a2 , . . .}) that is greater than or equal to
every element of {a1 , a2 , . . .}.12
11
12
For more details check http://en.wikipedia.org/wiki/Infimum
http://en.wikipedia.org/wiki/Supremum
35
• Limit inferior and limit superior of a sequence of real numbers
13
lim inf an = supk≥1 inf n≥k an
lim sup an = inf k≥1 supn≥k an .
Let A1 , A2 , . . . be a sequence of events. Then
• Limit inferior and limit superior of a sequence of sets
T
S+∞
lim sup An = +∞
k=1
n=k An
S
T+∞
lim inf An = +∞
k=1
n=k An .
•
Motivation 1.70 — A special case of the Fatou’s lemma
This result plays a vital role in the proof of continuity of probability functions.
•
Proposition 1.71 — A special case of the Fatou’s lemma (Resnick, 1999, p. 32)
Suppose A1 , A2 , . . . is a sequence of events in F. Then
P (lim inf An ) ≤ lim inf P (An ) ≤ lim sup P (An ) ≤ P (lim sup An ).
(1.50)
•
Exercise 1.72 — A special case of the Fatou’s lemma
Prove Proposition 1.71 (Resnick, 1999, pp. 32-33; Karr, 1993, p. 27).
•
Theorem 1.73 — Continuity (Karr, 1993, p. 27)
If An → A then P (An ) → P (A).
•
Exercise 1.74 — Continuity
Prove Theorem 1.73 by using Proposition 1.71 (Karr, 1993, p. 27).
•
Motivation 1.75 — (1st.) Borel-Cantelli Lemma (Resnick, 1999, p. 102)
This result is simple but still is a basic tool for proving almost sure convergence of
sequences of random variables.
•
13
http://en.wikipedia.org/wiki/Limit superior and limit inferior
36
Theorem 1.76 — (1st.) Borel-Cantelli Lemma (Resnick, 1999, p. 102)
Let A1 , A2 , . . . be any events in F. Then
+∞
X
P (An ) < +∞ ⇒ P (lim sup An ) = 0.
(1.51)
n=1
•
Exercise 1.77 — (1st.) Borel-Cantelli Lemma
Prove Theorem 1.76 (Resnick, 1999, p. 102).
37
•
1.4
Distribution
functions;
discrete,
absolutely
continuous and mixed probabilities
Motivation 1.78 — Distribution function (Karr, 1993, pp. 28-29)
A probability function P on the Borel σ−algebra B(IR) is determined by its values
P ((−∞, x]), for all intervals (−∞, x].
Probability functions on the real line play an important role as distribution functions
of random variables.
•
Definition 1.79 — Distribution function (Karr, 1993, p. 29)
Let P be a probability function defined on (IR, B(IR)). The distribution function
associated to P is represented by FP and defined by
FP (x) = P ((−∞, x]), x ∈ IR.
(1.52)
•
Theorem 1.80 — Some properties of distribution functions (Karr, 1993, pp. 2930)
Let FP be the distribution function associated to P . Then
1. FP is non-decreasing. Hence, the left-hand limit
FP (x− ) = lim FP (s)
(1.53)
s↑x, s<x
and the right-hand limit
FP (x+ ) = lim FP (s)
(1.54)
s↓x, s>x
exist for each x, and
FP (x− ) ≤ FP (x) ≤ FP (x+ ).
(1.55)
2. FP is right-continuous, i.e.
FP (x+ ) = FP (x)
(1.56)
for each x.
3. FP has the following limits:
lim FP (x) = 0
(1.57)
lim FP (x) = 1.
(1.58)
x→−∞
x→+∞
•
38
Definition 1.81 — Distribution function (Resnick, 1999, p. 33)
A function FP : IR → [0, 1] satisfying properties 1., 2. and 3. from Theorem 1.80 is called
a distribution function.
•
Exercise 1.82 — Some properties of distribution functions
Prove Theorem 1.80 (Karr, 1993, p. 30).
•
Definition 1.83 — Survival (or survivor) function (Karr, 1993, p. 31)
The survival (or survivor) function associated to P is
SP (x) = 1 − FP (x) = P ((x, +∞)), x ∈ IR.
(1.59)
•
SP (x) are also termed tail probabilities.
Proposition 1.84 — Probabilities in terms of the distribution function (Karr,
1993, p. 30)
The following table condenses the probabilities of various intervals in terms of the
distribution function
Interval I
Probability P (I)
(−∞, x]
FP (x)
(x, +∞)
1 − FP (x)
(−∞, x)
FP (x− )
[x, +∞)
1 − FP (x− )
(a, b]
FP (b) − FP (a)
[a, b)
FP (b− ) − FP (a− )
[a, b]
FP (b) − FP (a− )
(a, b)
FP (b− ) − FP (a)
{x}
FP (x) − FP (x− )
where x ∈ IR and −∞ < a < b < +∞.
•
39
Example 1.85 — Point mass (Karr, 1993, p. 31)
Let P be defined as
(
1 if s ∈ A
P (A) = s (A) =
0 otherwise,
(1.60)
for every event A ∈ F, i.e. P is a point mass at s. Then
FP (x) = P ((−∞, x])
(
0, x < s
=
1, x ≥ s.
(1.61)
•
Example 1.86 — Uniform distribution on [0, 1] (Karr, 1993, p. 31)
Let P be defined as
P ((a, b]) = Length((a, b] ∩ [0, 1]),
(1.62)
for any real interval (a, b] with −∞ < a < b < +∞. Then
FP (x) = P ((−∞, x])


 0, x < 0
=
x, 0 ≤ x ≤ 1


1, x > 1.
(1.63)
•
We are going to revisit the discrete and absolutely continuous probabilities and
introduce mixed distributions.
Definition 1.87 — Discrete probabilities (Karr, 1993, p. 32)
A probability P defined on (IR, B(IR)) is discrete if there is a countable set C such that
P (C) = 1.
•
Remark 1.88 — Discrete probabilities (Karr, 1993, p. 32)
Discrete probabilities are finite or countable convex combinations of point masses. The
associated distribution functions do not increase “smoothly” — they increase only by
means of jumps.
•
40
Proposition 1.89 — Discrete probabilities (Karr, 1993, p. 32)
Let P be a probability function on (IR, B(IR)). Then the following are equivalent:
1. P is a discrete probability.
2. There is a real sequence x1 , x2 , . . . and numbers p1 , p2 , . . . with pn > 0, for all n, and
P+∞
n=1 pn = 1 such that
P (A) =
+∞
X
pn × xn (A),
(1.64)
n=1
for all A ∈ B(IR).
3. There is a real sequence x1 , x2 , . . . and numbers p1 , p2 , . . . with pn > 0, for all n, and
P+∞
n=1 pn = 1 such that
FP (x) =
+∞
X
pn × 1(−∞,x] (xn ),
(1.65)
n=1
for all x ∈ IR.
•
Remark 1.90 — Discrete probabilities (Karr, 1993, p. 33)
The distribution function associated to a discrete probability increases only by jumps of
size pn at xn .
•
Exercise 1.91 — Discrete probabilities
Prove Proposition 1.89 (Karr, 1993, p. 32).
•
Example 1.92 — Discrete probabilities
Let px represent from now on P ({x}).
• Uniform distribution on a finite set C
px =
1
,
#C
P (A) =
x∈C
#A
,
#C
A ⊆ C.
This distribution is also known as the Laplace distribution.
• Bernoulli distribution with parameter p (p ∈ [0, 1])
C = {0, 1}
px = px (1 − p)1−x , x ∈ C.
This probability function arises in what we call a Bernoulli trial — a yes/no random
experiment which yields success with probability p.
41
• Binomial distribution with parameters n and p (n ∈ IN, p ∈ [0, 1])
C = {0, 1, . . . , n}
px = nx px (1 − p)n−x , x ∈ C.
The binomial distribution is the discrete probability distribution of the number of
successes in a sequence of n independent yes/no experiments, each of which yields
success with probability p.
Pn
= 1 follows from the binomial theorem
Moreover, the result
x=0 px
(http://en.wikipedia.org/wiki/Binomial theorem).
• Geometric distribution with parameter p (p ∈ [0, 1])
C = IN
px = (1 − p)x−1 p, x ∈ C.
This probability function is associated to the total number of
(i.i.d.)
Bernoulli trials needed to get one sucess — the first sucess
(http://en.wikipedia.org/wiki/Geometric distribution). The graph of
(
0,
x<1
FP (x) = P[x]
(1.66)
i−1
[x]
p = 1 − (1 − p) , x ≥ 1,
i=1 (1 − p)
where [x] represents the integer part of the real number x, follows:
42
• Negative binomial distribution with parameters r and p (r ∈ IN, p ∈ [0, 1])
C = {r, r + 1, . . .}
px = x−1
(1 − p)x−r pr , x ∈ C.
r−1
This probability function is associated to the total number of (i.i.d.)
Bernoulli trials needed to get a pre-specified number r of sucesses
The geometric
(http://en.wikipedia.org/wiki/Negative binomial distribution).
distribution is a particular case: r = 1.
• Hypergeometric
IN and M, n ≤ N )
distribution
with
parameters
N, M, n
(N, M, n
∈
C = {x ∈ IN0 : max{0, n − N + M } ≤ x ≤ min{n, M }}
−M
(Mx )(Nn−x
)
, x ∈ C.
(Nn )
It is a discrete probability that describes the number of successes in a
sequence of n draws from a finite population with size N without replacement
(http://en.wikipedia.org/wiki/Hypergeometric distribution).
px =
• Poisson distribution with parameter λ (λ ∈ IR+ )
C = IN0
x
px = e−λ λx! , x ∈ C.
It is discrete probability that expresses the probability of a number of events
occurring in a fixed period of time if these events occur with a known average rate
and independently of the time since the last event. The Poisson distribution can
also be used for the number of events in other specified intervals such as distance,
area or volume (http://en.wikipedia.org/wiki/Poisson distribution).
The figure comprises the distribution function for three different values of λ.
43
•
Motivation 1.93 — Absolutely continuous probabilities (Karr, 1993, p. 33)
Absolutely continuous probabilities are the antithesis of discrete probabilities in the sense
that they have “smooth” distribution functions.
•
Definition 1.94 — Absolutely continuous probabilities (Karr, 1993, p. 33)
A probability function P on (IR, B(IR)) is absolutely continuous if there is a non-negative
function fP on IR such that
Z b
P ((a, b]) =
fP (x)dx,
(1.67)
a
for every interval (a, b] ∈ B(IR).
•
Remark 1.95 — Absolutely continuous probabilities
If P is an absolutely continuous probability then FP (x) is an absolutely continuous real
function.
•
Remark 1.96 — Continuous, uniformly continuous and absolutely continuous
functions
• Continuous function
A real function f is continuous in x if for any sequence {x1 , x2 , . . .} such that
limn→∞ xn = x, it holds that limn→∞ f (xn ) = f (x).
One can say, briefly, that a function is continuous iff it preserves limits.
For the Cauchy definition (epsilon-delta) of
http://en.wikipedia.org/wiki/Continuous function
continuous
function
see
• Uniformly continuous function
Given metric spaces (X, d1 ) and (Y, d2 ), a function f : X → Y is called uniformly
continuous on X if for every real number > 0 there exists δ > 0 such that for
every x, y ∈ X with d1 (x, y) < δ, we have that d2 (f (x), f (y)) < .
If X and Y are subsets of the real numbers, d1 and d2 can be the standard Euclidean
norm, |.|, yielding the definition: for all > 0 there exists a δ > 0 such that for all
x, y ∈ X, |x − y| < δ implies |f (x) − f (y)| < .
The difference between being uniformly continuous, and simply being continuous at
every point, is that in uniform continuity the value of δ depends only on and not
on the point in the domain (http://en.wikipedia.org/wiki/Uniform continuity).
44
• Absolutely continuous function
Let (X, d) be a metric space and let I be an interval in the real line IR. A function f :
I → X is absolutely continuous on I if for every positive number , there is a positive
number δ such that whenever a (finite or infinite) sequence of pairwise disjoint subP
P
intervals [xk , yk ] of I satisfies k |yk − xk | < δ then k d (f (yk ), f (xk )) < .
Absolute continuity is a smoothness property which is stricter than continuity and
uniform continuity.
The two following functions are continuous everywhere but not absolutely
continuous:
(
0, if x = 0
1. f (x) =
x sin(1/x), if x 6= 0,
on a finite interval containing the origin;
2. f (x) = x2 on an unbounded interval.
(http://en.wikipedia.org/wiki/Absolute continuity)
•
Proposition 1.97 — Absolutely continuous probabilities (Karr, 1993, p. 34)
A probability function P is absolutely continuous iff there is a non-negative function f
on IR such that
Z +∞
f (s)ds = 1
(1.68)
−∞
Z x
FP (x) =
f (s)ds, x ∈ IR.
(1.69)
−∞
•
Exercise 1.98 — Absolutely continuous probabilities
Prove Proposition 1.97 (Karr, 1993, p. 34).
•
Example 1.99 — Absolutely continuous probabilities
• Uniform distribution on [a, b] (a, b ∈ IR, a < b)
(
1
, a≤x≤b
b−a
fP (x) =
0,
otherwise.


x<a
 0,
x−a
FP (x) =
, a≤x≤b
b−a


1,
x > b.
This absolutely continuous probability is such that all intervals of the same length
on the distribution’s support are equally probable.
45
The
support
is
defined
by
the
two
parameters,
a
and
b,
which
are
its
minimum
and
maximum
values
(http://en.wikipedia.org/wiki/Uniform distribution (continuous)).
• Exponential distribution with parameter λ (λ ∈ IR+ )
(
λe−λx , x ≥ 0
fP (x) =
0,
otherwise.
(
FP (x) =
0,
x<0
−λx
1 − e , x ≥ 0.
The
exponential
distribution
is
used
times
between
consecutive
events
in
a
(http://en.wikipedia.org/wiki/Exponential distribution).
14
to
describe
the
Poisson
process.14
I.e. a process in which events occur continuously and independently at a constant average rate.
46
Let P ∗ be the (discrete) Poisson probability with parameter λ x.
P ∗ ({0}) = e−λ x = P ((x, +∞)) = 1 − FP (x).
Then
• Normal distribution with parameters µ (µ ∈ IR) and σ 2 (σ 2 ∈ IR+ )
n
o
(x−µ)2
1
exp
−
, x ∈ IR
fP (x) = √2πσ
2
2σ 2
The normal distribution or Gaussian distribution is used describes data that cluster
around a mean or average. The graph of the associated probability density function
is bell-shaped, with a peak at the mean, and is known as the Gaussian function or
bell curve http://en.wikipedia.org/wiki/Normal distribution).
Rx
FP (x) = −∞ fP (s)ds, x ∈ IR
•
47
Motivation 1.100 — Mixed distributions (Karr, 1993, p. 34)
A probability function need not to be discrete or absolutely continuous...
•
Definition 1.101 — Mixed distributions (Karr, 1993, p. 34)
A probability function P is mixed if there is a discrete probability Pd , an absolutely
continuous probability Pc and α ∈ (0, 1) such that P is a convex combination of Pd and
Pc , that is,
P = αPd + (1 − α)Pc .
(1.70)
•
Example 1.102 — Mixed distributions
Let M (λ)/M (µ)/1 represent a queueing system with Poisson arrivals (rate λ > 0) and
exponential service times (rate µ > 0) and one server.
In the equilibrium state, the probability function associated to the waiting time of an
arriving customer is
P (A) = (1 − ρ){0} (A) + ρPExp(µ(1−ρ)) (A), A ∈ B(IR),
λ
µ
< 1 and
Z
µ(1 − ρ) e−µ(1−ρ) s ds.
PExp(µ(1−ρ)) (A) =
where 0 < ρ =
(1.71)
(1.72)
A
The associated distribution function is given by
(
0,
x<0
FP (x) =
−µ(1−ρ) x
(1 − ρ) + ρ 1 − e
, x ≥ 0.
(1.73)
•
48
1.5
Conditional probability
Motivation 1.103 — Conditional probability (Karr, 1993, p. 35)
We shall revise probabilities to account for the knowledge that an event has occurred,
using a concept known as conditional probability.
•
Definition 1.104 — Conditional probability (Karr, 1993, p. 35)
Let A and B be events. If P (A) > 0 the conditionally probability of B given A is equal to
P (B|A) =
P (B ∩ A)
.
P (A)
(1.74)
•
In case P (A) = 0, we make the convention that P (B|A) = P (B).
Remark 1.105 — Conditional probability (Karr, 1993, p. 35)
P (B|A) can be interpreted as the relative likelihood that B occurs given that A is known
to have occured.
•
Exercise 1.106 — Conditional probability
Solve exercises 1.23 and 1.24 of Karr (1993, p. 40).
•
Example 1.107 — Conditional probability (Grimmett and Stirzaker, 2001, p. 9)
A family has two children.
• What is the probability that both are boys, given that at least one is a boy?
The older and younger child may each be male or female, so there are four possible
combination of sexes, which we assume to be equally likely. Therefore
• Ω = {GG, GB, BG, BB}
where G = girl, B = boy, and P (GG) = P (GB) = P (BG) = P (BB) = 41 .
From the definition of conditional probability
P (BB|one boy at least) = P [BB|(GB ∪ BG ∪ BB)]
P [BB ∩ (GB ∪ BG ∪ BB)]
=
P (GB ∪ BG ∪ BB)
P (BB)
=
P (GB) + P (BG) + P (BB)
=
1
4
1
4
1
4
+ +
1
4
1
.
3
A popular but incorrect answer to this question is 12 .
=
49
(1.75)
This is the correct answer to another question:
• For a family with two children, what is the probability that both are boys given
that the younger is a boy?
In this case
P (BB|younger child is a boy) = P [BB|(GB ∪ BB)]
= ...
=
=
=
P (BB)
P (GB) + P (BB)
1
4
1
4
+
1
4
1
.
2
(1.76)
•
Exercise 1.108 — Conditional probability (Grimmett and Stirzaker, 2001, p. 9)
The prosecutor’s fallacy15 — Let G be the event that an accused is guilty, and T the event
that some testimony is true.
Some lawyers have argued on the assumption that P (G|T ) = P (T |G). Show that this
holds iff P (G) = P (T ).
•
Motivation 1.109 — Multiplication rule (Montgomery and Runger, 2003, p. 42)
The definition of conditional probability can be rewritten to provide a general expression
for the probability of the intersection of (two) events. This formula is referred to as a
multiplication rule for probabilities.
•
Proposition 1.110 — Multiplication rule (Montgomery and Runger, 2003, p. 43)
Let A and B be two events. Then
P (A ∩ B) = P (B|A) × P (A) = P (A|B) × P (B).
(1.77)
More generally: let A1 , . . . , An be events then
P (A1 ∩ A2 ∩ . . . ∩ An−1 ∩ An ) = P (A1 ) × P (A2 |A1 ) × P [A3 |(A1 ∩ A2 )]
. . . × P [An |(A1 ∩ A2 ∩ . . . An−1 )].
(1.78)
•
15
The
prosecution
made
this
error
(http://en.wikipedia.org/wiki/Alfred Dreyfus) in 1894.
50
in
the
famous
Dreyfus
affair
Example 1.111 — Multiplication rule (Montgomery and Runger, 2003, p. 43)
The probability that an automobile battery, subject to high engine compartment
temperature, suffers low charging current is 0.7. The probability that a battery is subject
to high engine compartment temperature 0.05.
What is the probability that a battery suffers low charging current and is subject to
high engine compartment temperature?
• Table of events and probabilities
Event
Probability
C = battery suffers low charging current
P (C) =?
T = battery subject to high engine compartment temperature
P (T ) = 0.05
C|T = battery suffers low charging current given that it is
P (C|T ) = 0.7
subject to high engine compartment temperature
• Probability
P (C ∩ T )
mult. rule
=
P (C|T ) × P (T )
=
0.7 × 0.05
=
0.035.
(1.79)
•
Motivation 1.112 — Law of total probability (Karr, 1993, p. 35)
The next law expresses the probability of an event in terms of its conditional probabilities
given elements of a partition of Ω.
•
Proposition 1.113 — Law of total probability (Karr, 1993, p. 35)
Let {A1 , A2 , . . .} a countable partition of Ω. Then, for each event B,
P (B) =
+∞
X
P (B|Ai ) × P (Ai ).
(1.80)
i=1
•
Exercise 1.114 — Law of total probability
Prove Proposition 1.113 by using σ−additivity of a probability function and the fact that
S
B = +∞
•
i=1 (B ∩ Ai ) (Karr, 1993, p. 36).
51
Corollary 1.115 — Law of total probability (Montgomery and Runger, 2003, p. 44)
For any events A and B,
P (B) = P (B|A) × P (A) + P (B|Ac ) × P (Ac ).
(1.81)
•
Example 1.116 — Law of total probability (Grimmett and Stirzaker, 2001, p. 11)
Only two factories manufacture zoggles. 20% of the zoggles from factory I and 5% from
factory II are defective. Factory I produces twice as many zoggles as factory II each week.
What is the probability that a zoggle, randomly chosen from a week’s production, is
not defective?
• Table of events and probabilities
Event
Probability
D = defective zoggle
P (D) =?
A = zoggle made in factory I
P (A) = 2 × [1 − P (A)] =
Ac = zoggle made in factory II
P (Ac ) = 1 − P (A) =
D|A = defective zoggle given that it is made in factory I
P (D|A) = 0.20
D|Ac = defective zoggle given that it is made in factory II
P (D|Ac ) = 0.05
2
3
1
3
• Probability
P (Dc )
=
lawtotalprob
=
=
=
1 − P (D)
1 − [P (D|A) × P (A) + P (D|Ac ) × P (Ac )]
2
1
1 − 0.20 × + 0.05 ×
3
3
51
.
60
•
52
Motivation 1.117 — Bayes’ theorem (Karr, 1993, p. 36)
Traditionally (and probably incorrectly) attributed to the English cleric Thomas Bayes
(http://en.wikipedia.org/wiki/Thomas Bayes), the theorem that bears his name is used
to compute conditional probabilities “the other way around”.
•
Proposition 1.118 — Bayes’ theorem (Karr, 1993, p. 36)
Let {A1 , A2 , . . .} be a countable partition of Ω. Then, for each event B with P (B) > 0
and each n,
P (B|An )P (An )
P (B)
P (B|An )P (An )
= P+∞
i=1 P (B|Ai )P (Ai )
P (An |B) =
(1.82)
•
Exercise 1.119 — Bayes’ theorem (Karr, 1993, p. 36)
Prove Bayes’ theorem by using the definition of conditional probability and the law of
total probability (Karr, 1993, p. 36).
•
Example 1.120 — Bayes’ theorem (Grimmett and Stirzaker, 2003, p. 11)
Resume Example 1.116...
If the chosen zoggle is defective, what is the probability that it came from factory II.
• Probability
P (D|A) × P (A)
P (D)
0.20 × 23
=
1 − 51
60
8
=
.
9
P (A|D) =
(1.83)
•
53
References
• Grimmett, G.R. and Stirzaker, D.R. (2001). Probability and Random Processes (3rd.
edition). Oxford. (QA274.12-.76.GRI.30385 and QA274.12-.76.GRI.40695 refer to
the library code of the 1st. and 2nd. editions from 1982 and 1992, respectively.)
• Karr, A.F. (1993). Probability. Springer-Verlag.
• Montgomery, D.C. and Runger, G.C. (2003). Applied statistics and probability for
engineers. John Wiley & Sons, New York. (QA273-280/3.MON.64193)
• Papoulis, A. (1965). Probability, Random Variables and Stochastic Processes.
McGraw-Hill Kogakusha, Ltd. (QA274.12-.76.PAP.28598)
• Resnick, S.I. (1999). A Probability Path. Birkhäuser. (QA273.4-.67.RES.49925)
• Righter, R. (200–). Lectures notes for the course Probability and Risk Analysis
for Engineers. Department of Industrial Engineering and Operations Research,
University of California at Berkeley.
• Yates, R.D. and Goodman, D.J. (1999). Probability and Stochastic Processes: A
friendly Introduction for Electrical and Computer Engineers. John Wiley & Sons,
Inc. (QA273-280/4.YAT.49920)
54
Chapter 2
Random variables
2.1
Fundamentals
Motivation 2.1 — Inverse image of sets (Karr, 1993, p. 43)
Before we introduce the concept of random variable (r.v.) we have to talk rather
extensively on inverse images of sets and inverse image mapping.
•
Definition 2.2 — Inverse image (Karr, 1993, p. 43)
Let:
• X be a function with domain Ω and range Ω0 , i.e. X : Ω → Ω0 ;
• F and F 0 be the σ − algebras on Ω and Ω0 , respectively.
(Frequently Ω0 = IR and F 0 = B(IR).) Then the inverse image under X of the set B ∈ F 0
is the subset of Ω given by
X −1 (B) = {ω : X(ω) ∈ B},
(2.1)
written from now on {X ∈ B} (graph!).
•
Remark 2.3 — Inverse image mapping (Karr, 1993, p. 43)
The inverse image mapping X −1 maps subsets of Ω0 to subsets of Ω. X −1 preserves all
set operations, as well as disjointness.
•
55
Proposition 2.4 — Properties of inverse image mapping (Karr, 1993, p. 43;
Resnick, 1999, p. 72)
Let:
• X : Ω → Ω0 ;
• F and F 0 be the σ − algebras on Ω and Ω0 , respectively;
• B, B 0 and {Bi : i ∈ I} be sets in F 0 .
Then:
1. X −1 (∅) = ∅
2. X −1 (Ω0 ) = Ω
3. B ⊆ B 0 ⇒ X −1 (B) ⊆ X −1 (B 0 )
S
S
4. X −1 ( i∈I Bi ) = i∈I X −1 (Bi )
T
T
5. X −1 ( i∈I Bi ) = i∈I X −1 (Bi )
6. B ∩ B 0 = ∅ ⇒ X −1 (B) ∩ X −1 (B 0 ) = ∅
7. X −1 (B c ) = [X −1 (B)]c .
•
Exercise 2.5 — Properties of inverse image mapping
Prove Proposition 2.4 (Karr, 1993, p. 43).
•
Proposition 2.6 — σ − algebras and inverse image mapping (Resnick, 1999, pp.
72–73)
Let X : Ω → Ω0 be a mapping with inverse image. If F 0 is a σ − algebra on Ω0 then
X −1 (F 0 ) = {X −1 (B) : B ∈ F 0 }
(2.2)
is a σ − algebra on Ω.
Exercise 2.7 — σ − algebras and inverse image mapping
Prove Proposition 2.6 by verifying the 3 postulates for a σ − algebra (Resnick, 1999, p.
73).
•
56
Proposition 2.8 — Inverse images of σ − algebras generated by classes of
subsets (Resnick, 1999, p. 73)
Let C 0 be a class of subsets of Ω0 . Then
X −1 (σ(C 0 )) = σ({X −1 (C 0 )}),
(2.3)
i.e., the inverse image of the σ − algebra generated by C 0 is the same as the σ − algebra
on Ω generated by the inverse images.
•
Exercise 2.9 — Inverse images of σ − algebras generated by classes of subsets
Prove Proposition 2.8. This proof comprises the verification of the 3 postulates for a
σ − algebra (Resnick, 1999, pp. 73–74) and much more.
•
Definition 2.10 — Measurable space (Resnick, 1999, p. 74)
The pair (Ω, F) consisting of a set Ω and a σ − algebra on Ω is called a measurable
space.
•
Definition 2.11 — Measurable map (Resnick, 1999, p. 74)
Let (Ω, F) and (Ω0 , F 0 ) be two measurable spaces. Then a map X : Ω → Ω0 is called a
measurable map if
X −1 (F 0 ) ⊆ F.
(2.4)
•
Remark 2.12 — Measurable maps/ Random variables (Karr, 1993, p. 44)
A special case occurs when (Ω0 , F 0 ) = (IR, B(IR)) — in this case X is called a random
variable. That is, random variables are functions on the sample space Ω for which inverse
images of Borel sets are events of Ω.
•
Definition 2.13 — Random variable (Karr, 1993, p. 44)
Let (Ω, F) and (Ω0 , F 0 ) = (IR, B(IR)) be two measurable spaces. A random variable (r.v.)
is a function X : Ω → IR such that
X −1 (B) ∈ F, ∀B ∈ B(IR).
(2.5)
•
57
Remark 2.14 — Random variables (Karr, 1993, p. 44)
A r.v. is a function on the sample space: it transforms events into real sets.
The technical requirement that sets {X ∈ B} = X −1 (B) be events of Ω is needed in
order that the probability
P ({X ∈ B}) = P (X −1 (B))
(2.6)
•
be defined.
Motivation 2.15 — Checking if X is a r.v. (Karr, 1993, p. 47)
To verify that X is a r.v. it is not necessary to check that {X ∈ B} = X −1 (B) ∈ F for all
Borel sets B. In fact, σ(X) ⊆ F once X −1 (B) ∈ F for enough “elementary” Borel sets.
•
Proposition 2.16 — Checking if X is a r.v. (Resnick, 1999, p. 77; Karr, 1993, p. 47)
The real function X : Ω → IR is a r.v. iff
{X ≤ x} = X −1 ((−∞, x]) ∈ F, ∀x ∈ IR.
(2.7)
Similarly if we replace {X ≤ x} by {X > x}, {X < x} or {X ≥ x}.
Example 2.17 — Random variable
• Random experiment
Throw a traditional fair die and observe the number of points.
• Sample space
Ω = {1, 2, 3, 4, 5, 6}
• σ−algebra on Ω
Let us consider a non trivial one:
F = {∅, {1, 3, 5}, {2, 4, 6}, Ω}
• Random variable
X : Ω → IR such that: X(1) = X(3) = X(5) = 0 and X(2) = X(4) = X(6) = 1
58
•
• Inverse image mapping
Let B ∈ B(IR). Then

∅,



 {1, 3, 5},
X −1 (B) =

{2, 4, 6},



Ω,
if
if
if
if
0 6∈ B, 1 6∈ B
0 ∈ B, 1 6∈ B
0 6∈ B, 1 ∈ B
0 ∈ B, 1 ∈ B
∈ F, ∀B ∈ B(IR).
(2.8)
Therefore X is a r.v. defined in F.
• A function which is not a r.v.
Y : Ω → IR such that: Y (1) = Y (2) = Y (3) = 1 and Y (4) = Y (5) = Y (6) = 0.
Y is not a r.v. defined in F because Y −1 ({1}) = {1, 2, 3} 6∈ F.
•
There are generalizations of r.v.
Definition 2.18 — Random vector (Karr, 1993, p. 45)
A d − dimensional random vector is a function X = (X1 , . . . , Xd ) : Ω → IRd such that
each component Xi , i = 1, . . . , d, is a random variable.
•
Remark 2.19 — Random vector (Karr, 1993, p. 45)
Random vectors will sometimes be treated as finite sequences of random variables.
•
Definition 2.20 — Stochastic process (Karr, 1993, p. 45)
A stochastic process with index set (or parameter space) T is a collection {Xt : t ∈ T } of
r.v. (indexed by T ).
•
Remark 2.21 — Stochastic process (Karr, 1993, p. 45)
Typically:
• T = IN0 and {Xn : n ∈ IN0 } is called a discrete time stochastic process;
• T = IR0+ and {Xt : t ∈ IR0+ } is called a continuous time stochastic process.
59
•
Proposition 2.22 — σ−algebra generated by a r.v. (Karr, 1993, p. 46)
The family of events that are inverse images of Borel sets under a r.v is a σ − algebra on
Ω. In fact, given a r.v. X, the family
σ(X) = {X −1 (B) : B ∈ B(IR)}
(2.9)
is a σ − algebra on Ω, known as the σ − algebra generated by X.
•
Remark 2.23 — σ−algebra generated by a r.v.
• Proposition 2.22 is a particular case of Proposition 2.6 when F 0 = B(IR).
• Moreover, σ(X) is a σ − algebra for every function X : Ω → IR; and X is a r.v. iff
σ(X) ⊆ F, i.e., iff X is a measurable map (Karr, 1993, p. 46).
•
Example 2.24 — σ−algebra generated by an indicator r.v. (Karr, 1993, p. 46)
Let:
• A be a subset of the sample space Ω;
• X : Ω → IR such that
(
X(ω) = 1A (w) =
1, ω ∈ A
0, ω ∈
6 A.
(2.10)
Then X is the indicator r.v. of an event A. In addition,
σ(X) = σ(1A ) = {∅, A, Ac , Ω}
(2.11)

∅,



 Ac ,
X −1 (B) =

A,



Ω,
(2.12)
since
if
if
if
if
0 6∈ B, 1 6∈ B
0 ∈ B, 1 6∈ B
0 6∈ B, 1 ∈ B
0 ∈ B, 1 ∈ B,
for any B ∈ B(IR).
•
Example 2.25 — σ−algebra generated by a constant r.v.
Let X : Ω → IR such that X(ω) = c, ∀ω ∈ Ω. Then
(
∅, if c 6∈ B
X −1 (B) =
Ω, if c ∈ B,
for any B ∈ B(IR), and σ(X) = {∅, Ω} (trivial σ − algebra).
60
(2.13)
•
Example 2.26 — σ−algebra generated by a simple r.v. (Karr, 1993, pp. 45-46)
A simple r.v. takes only finitely many values and has the form
X=
n
X
ai × 1Ai ,
(2.14)
i=1
where ai , i = 1, . . . , n, are (not necessarily distinct) real numbers and Ai , i = 1, . . . , n,
are events that constitute a partition of Ω. X is a r.v. since
{X ∈ B} =
n
[
{Ai : ai ∈ B},
(2.15)
i=1
for any B ∈ B(IR).
For this simple r.v. we get
σ(X) = σ({A1 , . . . , An })
[
= { Ai : I ⊆ {1, . . . , n}},
(2.16)
i∈I
•
regardless of the values of a1 , . . . , an .
Definition 2.27 — σ−algebra generated by a random vector (Karr, 1993, p. 46)
The σ−algebra generated by the d − dimensional random vector (X1 , . . . , Xd ) : Ω → IRd
is given by
σ((X1 , . . . , Xd )) = {(X1 , . . . , Xd )−1 (B) : B ∈ B(IRd )}.
(2.17)
•
61
2.2
Combining random variables
To work with r.v., we need assurance that algebraic, limiting and transformation
operations applied to them yield other r.v.
In the next proposition we state that the set of r.v. is closed under:
• addition and scalar multiplication;1
• maximum and minimum;
• multiplication;
• division.
Proposition 2.28 — Closure under algebraic operations (Karr, 1993, p. 47)
Let X and Y be r.v. Then:
1. aX + bY is a r.v., for all a, b ∈ IR;
2. max{X, Y } and min{X, Y } are r.v.;
3. XY is a r.v.;
X
Y
is a r.v. provided that Y (ω) 6= 0, ∀ω ∈ Ω.
•
Exercise 2.29 — Closure under algebraic operations
Prove Proposition 2.28 (Karr, 1993, pp. 47–48).
•
4.
Corollary 2.30 — Closure under algebraic operations (Karr, 1993, pp. 48–49)
Let X : Ω → IR be a r.v. Then
X + = max{X, 0}
X
−
(2.18)
= − min{X, 0},
(2.19)
the positive and negative parts of X (respectively), are non-negative r.v., and so is
|X| = X + + X −
(2.20)
•
is a r.v.
1
I.e. the set of r.v. is a vector space.
62
Remark 2.31 — Canonical representation of a r.v. (Karr, 1993, p. 49)
A r.v. can be written as a difference of its positive and negative parts:
X = X + − X −.
(2.21)
•
Theorem 2.32 — Closure under limiting operations (Karr, 1993, p. 49)
Let X1 , X2 , . . . be r.v. Then sup Xn , inf Xn , lim sup Xn and lim inf Xn are r.v.
Consequently if
X(ω) = lim Xn (ω)
(2.22)
n→+∞
exists for every ω ∈ Ω, then X is also a r.v.
•
Exercise 2.33 — Closure under limiting operations
Prove Theorem 2.32 by noting that
{sup Xn ≤ x} = (sup Xn )−1 ((−∞, x])
+∞
\
{Xn ≤ x}
=
=
n=1
+∞
\
(Xn )−1 ((−∞, x])
(2.23)
n=1
{inf Xn ≥ x} = (inf Xn )−1 ([x, +∞))
+∞
\
=
{Xn ≥ x}
=
n=1
+∞
\
(Xn )−1 ([x, +∞))
(2.24)
n=1
lim sup Xn = inf sup Xn
(2.25)
lim inf Xn = sup inf Xn
(2.26)
k m≥k
k
m≥k
and that when X = limn→+∞ Xn exists, X = lim sup Xn = lim inf Xn (Karr, 1993,
p. 49).
•
Corollary 2.34 — Series of r.v. (Karr, 1993, p. 49)
P
If X1 , X2 , . . . are r.v., then provided that X(ω) = +∞
n=1 Xn (ω) converges for each ω, X is
a r.v.
•
63
Motivation 2.35 — Transformations of r.v. and random vectors (Karr, 1993, p.
50)
Another way of constructing r.v. is as functions of other r.v.
•
Definition 2.36 — Borel measurable function (Karr, 1993, p. 66)
A function g : IRn → IRm (for fixed n, m ∈ IN ) is a Borel measurable if
g −1 (B) ∈ B(IRn ), ∀B ∈ B(IRm ).
(2.27)
•
Remark 2.37 — Borel measurable function (Karr, 1993, p. 66)
• In order that g : IRn → IR be Borel measurable it suffices that
g −1 ((−∞, x]) ∈ B(IRn ), ∀x ∈ IR.
(2.28)
• A function g : IRn → IRm Borel measurable iff each of its components is Borel
measurable as a function from IRn to IR.
• Indicator functions, monotone functions and continuous functions are Borel
measurable.
• Moreover, the class of Borel measurable function has the closure properties under
algebraic and limiting operations as the family of r.v. on a probability space
(Ω, F, P ).
•
Theorem 2.38 — Transformations of random vectors (Karr, 1993, p. 50)
Let:
• X1 , . . . , Xd be r.v.;
• g : IRd → IR be a Borel measurable function.
Then Y = g(X1 , . . . , Xd ) is a r.v.
•
Exercise 2.39 — Transformations of r.v.
Prove Theorem 2.38 (Karr, 1993, p. 50).
•
Corollary 2.40 — Transformations of r.v. (Karr, 1993, p. 50)
Let:
• X be r.v.;
• g : IR → IR be a Borel measurable function.
•
Then Y = g(X) is a r.v.
64
2.3
Distributions and distribution functions
The main importance of probability functions on IR is that they are distributions of r.v.
Proposition 2.41 — R.v. and probabilities on IR (Karr, 1993, p. 52)
Let X be r.v. Then the set function
PX (B) = P (X −1 (B)) = P ({X ∈ B})
(2.29)
•
is a probability function on IR.
Exercise 2.42 — R.v. and probabilities on IR
Prove Proposition 2.41 by checking if the three axioms in the definition of probability
function hold (Karr, 1993, p. 52).
•
Definition 2.43 — Distribution, distribution and survival function of a r.v.
(Karr, 1993, p. 52)
Let X be a r.v. Then
1. the probability function on IR
PX (B) = P (X −1 (B)) = P ({X ∈ B}), B ∈ B(IR), is the distribution of X;
2. FX (x) = PX ((−∞, x]) = P (X −1 ((−∞, x]) = P ({X ≤ x}), x ∈ IR, is the
distribution function of X;
3. SX (x) = 1 − FX (x) = PX ((x, +∞)) = P (X −1 ((x, +∞)) = P ({X > x}), x ∈ IR, is
the survival (or survivor) function of X.
•
Definition 2.44 — Discrete and absolutely continuous r.v. (Karr, 1993, p. 52)
X is said to be a discrete (resp. absolutely continuous) r.v. if PX is a discrete (resp.
absolutely continuous) probability function.
•
Motivation 2.45 — Confronting r.v.
How can we confront two r.v. X and Y ?
•
65
Definition 2.46 — Identically distributed r.v. (Karr, 1993, p. 52)
Let X and Y be two r.v. Then X and Y are said to be identically distributed — written
d
X = Y — if
PX (B) = P ({X ∈ B})
= P ({Y ∈ B}) = PY (B), B ∈ B(IR),
i.e. if FX (x) = P ({X ≤ x) = P ({Y ≤ x}) = FY (x), x ∈ IR.
(2.30)
•
Definition 2.47 — Equal r.v. almost surely (Karr, 1993, p. 52; Resnick, 1999, p.
167)
a.s.
Let X and Y be two r.v. Then X is equal to Y almost surely — written X = Y — if
P ({X = Y }) = 1.
(2.31)
•
Remark 2.48 — Identically distributed r.v. vs. equal r.v. almost surely (Karr,
1993, p. 52)
Equality in distribution of X and Y has no bearing on their equality as functions on Ω,
i.e.
d
a.s.
(2.32)
d
(2.33)
X = Y 6⇒ X = Y,
even though
a.s.
X = Y ⇒ X = Y.
•
Example 2.49 — Identically distributed r.v. vs. equal r.v. almost surely
• X ∼ Bernoulli(0.5)
P ({X = 0}) = P ({X = 1}) = 0.5
• Y = 1 − X ∼ Bernoulli(0.5) since
P ({Y = 0}) = P ({1 − X = 0}) = P ({X = 1}) = 0.5
P ({Y = 1}) = P ({1 − X = 1}) = P ({X = 0}) = 0.5
d
a.s.
• X = Y but X 6= Y .
•
66
Exercise 2.50 — Identically distributed r.v. vs. equal r.v. almost surely
a.s.
d
Prove that X = Y ⇒ X = Y .
•
Definition 2.51 — Distribution and distribution function of a random vector
(Karr, 1993, p. 53)
Let X = (X1 , . . . , Xd ) be a d − dimensional random vector. Then
1. the probability function on IRd
PX (B) = P (X −1 (B)) = P ({X ∈ B}), B ∈ B(IRd ), is the distribution of X;
2. the distribution function of X = (X1 , . . . , Xd ), also known as the joint distribution
function of X1 , . . . , Xd is the function FX : IRd → [0, 1] given by
FX (x) = F(X1 ,...,Xd ) (x1 , . . . , xd )
= P ({X1 ≤ x1 , . . . , Xd ≤ xd }),
for any x = (x1 , . . . , xd ) ∈ IRd .
(2.34)
•
Remark 2.52 — Distribution function of a random vector (Karr, 1993, p. 53)
The distribution PX is determined uniquely by FX .
•
Motivation 2.53 — Marginal distribution function (Karr, 1993, p. 53)
Can we obtain the distribution of Xi from the joint distribution function?
•
Proposition 2.54 — Marginal distribution function (Karr, 1993, p. 53)
Let X = (X1 , . . . , Xd ) be a d − dimensional random vector. Then, for each i (i = 1, . . . , d)
and x (x ∈ IR),
FXi (x) =
lim
xj →+∞,j6=i
F(X1 ,...,Xi−1 ,Xi ,Xi+1 ,...,Xd ) (x1 , . . . , xi−1 , x, xi+1 , . . . , xd ).
(2.35)
•
Exercise 2.55 — Marginal distribution function
Prove Proposition 2.54 by noting that {X1 ≤ x1 , . . . , Xi−1 ≤ xi−1 , Xi ≤ x, Xi+1 ≤ xi+1 ,
. . . , Xd ≤ xd } ↑ {Xi ≤ x} when xj → +∞, j 6= i, and by considering the monotone
continuity of probability functions (Karr, 1993, p. 53).
•
67
Definition 2.56 — Discrete random vector (Karr, 1993, pp. 53–54)
The random vector X = (X1 , . . . , Xd ) is said to be discrete if X1 , . . . , Xd are discrete r.v.
i.e. if there is a countable set C ⊂ IRd such that P ({X ∈ C}) = 1.
•
Definition 2.57 — Absolutely continuous random vector (Karr, 1993, pp. 53–54)
The random vector X = (X1 , . . . , Xd ) is absolutely continuous if there is a non-negative
function fX : IRd → IR0+ such that
Z xd
Z x1
fX (s1 , . . . , sd ) dsd . . . ds1 ,
...
FX (x) =
(2.36)
−∞
−∞
for every x = (x1 , . . . , xd ) ∈ IRd . fX is called the joint density function (of X1 , . . . , Xd ). •
Proposition 2.58 — Absolutely continuous random vector; marginal density
function (Karr, 1993, p. 54)
If X = (X1 , . . . , Xd ) is absolutely continuous then, for each i (i = 1, . . . , d), Xi is
absolutely continuous and
Z x1
Z xd
...
fX (s1 , . . . , si−1 , x, si+1 , . . . , sd ) dsd . . . dsi−1 dsi+1 . . . ds1 . (2.37)
fXi (x) =
−∞
−∞
fXi is termed the marginal density function of Xi .
•
Remark 2.59 — Absolutely continuous random vector (Karr, 1993, p. 54)
If the random vector is absolutely continuous then any “sub-vector” is absolutely
continuous. Moreover, the converse of Proposition 2.58 is not true, that is, the fact that
X1 , . . . , Xd are absolutely continuous does not imply that (X1 , . . . , Xd ) is an absolutely
continuous random vector.
•
68
2.4
Key r.v. and random vectors and distributions
2.4.1
Discrete r.v. and random vectors
Integer-valued r.v. like the Bernoulli, binomial, hypergeometric, geometric, negative
binomial, hypergeometric and Poisson, and integer-valued random vectors like the
multinomial are discrete r.v. and random vectors of great interest.
• Uniform distribution on a finite set
Notation
X ∼ Uniform({x1 , x2 , . . . , xn })
Parameter
{x1 , x2 , . . . , xn } (xi ∈ IR, i = 1, . . . , n)
Range
{x1 , x2 , . . . , xn }
P.f.
P ({X = x}) = n1 , x = x1 , x2 , . . . , xn
This simple r.v. has the form X =
Pn
i=1
xi × 1{xi } .
• Bernoulli distribution
Notation
X ∼ Bernoulli(p)
Parameter
p = P (sucess) (p ∈ [0, 1])
Range
{0, 1}
P.f.
P ({X = x}) = px (1 − p)1−x , x = 0, 1
A Bernoulli distributed r.v. X is the indicator function of the event {X = 1}.
• Binomial distribution
Notation
X ∼ Binomial(n, p)
Parameters
n = number of Bernoulli trials (n ∈ IN )
p = P (sucess) (p ∈ [0, 1])
Range
{0, 1, . . . , n}
P.f.
P ({X = x}) =
n
x
px (1 − p)n−x , x = 0, 1, . . . , n
The binomial r.v. results from the sum of n i.i.d. Bernoulli distributed r.v.
69
• Geometric distribution
Notation
X ∼ Geometric(p)
Parameter
p = P (sucess) (p ∈ [0, 1])
Range
IN = {1, 2, 3, . . .}
P.f.
P ({X = x}) = (1 − p)x−1 p, x = 1, 2, 3, . . .
This r.v. satisfies the lack of memory property:
P ({X > k + x}|{X > k}) = P ({X > x}), ∀k, x ∈ IN.
(2.38)
• Negative binomial distribution
Notation
X ∼ NegativeBinomial(r, p)
Parameters
r = pre-specified number of sucesses (r ∈ IN )
p = P (sucess) (p ∈ [0, 1])
Range
{r, r + 1, . . .}
P.f.
P ({X = x}) =
x−1
r−1
(1 − p)x−r pr , x = r, r + 1, . . .
The negative binomial r.v. results from the sum of r i.i.d. geometrically distributed
r.v.
• Hypergeometric distribution
Notation
X ∼ Hypergeometric(N, M, n)
Parameters
N = population size (N ∈ IN )
M = sub-population size (M ∈ IN, M ≤ N )
n = sample size (n ∈ IN, n ≤ N )
Range
{max{0, n − N + M }, . . . , min{n, M }}
(M )(N −M )
P ({X = x}) = x Nn−x , x = max{0, n − N + M }, . . . , min{n, M }
(n)
P.f.
Note that the sample is collected without replacement.
).
Binomial(n, M
N
70
Otherwise X
∼
• Poisson distribution
Notation
X ∼ Poisson(λ)
Parameter
λ (λ ∈ IR+ )
Range
IN0 = {0, 1, 2, 3, . . .}
P.f.
P ({X = x}) = e−λ λx! , x = 0, 1, 2, 3, . . .
x
The distribution was proposed by Siméon-Denis Poisson (1781–1840) and published,
together with his probability theory, in 1838 in his work Recherches sur la probabilité
des jugements en matiéres criminelles et matiére civile (Research on the probability
of judgments in criminal and civil matters). The Poisson distribution can be derived
as a limiting case of the binomial distribution.2
In 1898 Ladislaus Josephovich Bortkiewicz (1868–1931) published a book titled The
Law of Small Numbers. In this book he first noted that events with low frequency
in a large population follow a Poisson distribution even when the probabilities of
the events varied. It was that book that made the Prussian horse-kick data famous.
Some historians of mathematics have even argued that the Poisson distribution
should have been named the Bortkiewicz distribution.3
• Multinomial distribution
In probability theory, the multinomial distribution is a generalization of the binomial
distribution when we are dealing not only with two types of events — a success with
probability p and a failure with probability 1 − p — but with d types of events with
P
probabilities p1 , . . . , pd such that p1 , . . . , pd ≥ 0, di=1 pi = 1.4
Notation
X = (X1 , . . . , Xd ) ∼ Multinomiald−1 (n, (p1 , . . . , pd ))
Parameters
n = number of Bernoulli trials (n ∈ IN )
(p1 , . . . , pd ) where pi = P (event of type i)
P
(p1 , . . . , pd ≥ 0, di=1 pi = 1)
P
{(n1 , . . . , nd ) ∈ IN0d : di=1 ni = n}
Qd
ni
P ({X1 = n1 , . . . , Xd = nd }) = Qd n!
i=1 pi ,
i=1 ni !
P
(n1 , . . . , nd ) ∈ IN0d : di=1 ni = n
Range
P.f.
2
http://en.wikipedia.org/wiki/Poisson distribution
http://en.wikipedia.org/wiki/Ladislaus Bortkiewicz
4
http://en.wikipedia.org/wiki/Multinomial distribution
3
71
Exercise 2.60 — Binomial r.v. (Grimmett and Stirzaker, 2001, p. 25)
DNA fingerprinting — In a certain style of detective fiction, the sleuth is required to
declare the criminal has the unusual characteristics...; find this person you have your
man. Assume that any given individual has these unusual characteristics with probability
10−7 (independently of all other individuals), and the city in question has 107 inhabitants.
Given that the police inspector finds such person, what is the probability that there
is at least one other?
•
Exercise 2.61 — Binomial r.v. (Righter, 200–)
A student (Fred) is getting ready to take an important oral exam and is concerned about
the possibility of having an on day or an off day. He figures that if he has an on day,
then each of his examiners will pass him independently of each other, with probability
0.8, whereas, if he has an off day, this probability will be reduced to 0.4.
Suppose the student will pass if a majority of examiners pass him. If the student feels
that he is twice as likely to have an off day as he is to have an on day, should he request
an examination with 3 examiners or with 5 examiners?
•
Exercise 2.62 — Geometric r.v.
Prove that the distribution function of X ∼ Geometric(p) is given by
(
0,
x<1
FX (x) = P (X ≤ x) = P[x]
i−1
[x]
p = 1 − (1 − p) , x ≥ 1,
i=1 (1 − p)
(2.39)
•
where [x] represents the integer part of x.
Exercise 2.63 — Hypergeometric r.v. (Righter, 200–)
From a mix of 50 widgets from supplier 1 and 100 from supplier 2, 10 widgets are randomly
selected and shipped to a customer.
What is the probability that all 10 came from supplier 1?
•
Exercise 2.64 — Poisson r.v. (Grimmett and Stirzaker, 2001, p. 19)
In your pocket is a random number N of coins, where N ∼ Poisson(λ). You toss each
coin once, with heads showing with probability p each time.
Show that the total number of heads has a Poisson distribution with parameter λp. •
72
Exercise 2.65 — Negative hypergeometric r.v. (Grimmett and Stirzaker, 2001, p.
19)
Capture-recapture — A population of N animals has had a number M of its members
captured, marked, and released. Let X be the number of animals it is necessary to
recapture (without re-release) in order to obtain r marked animals.
Show that
M M −1 N −M
P ({X = x}) =
N
r−1
x−r
N −1
x−1
.
(2.40)
•
Exercise 2.66 — Discrete random vectors
Prove that if
• Y ∼ Poisson(λ)
• (X1 , . . . , Xd )|{Y = n} ∼ Multinomiald−1 (n, (p1 , . . . , pd ))
then Xi ∼ Poisson(λpi ), i = 1, . . . , d.
•
Exercise 2.67 — Relating the p.f. of the negative binomial and binomial r.v.
Let X ∼ NegativeBinomial(r, p) and Y ∼ Binomial(x − 1, p). Prove that, for x = r, r +
1, r + 2, . . . and r = 1, 2, 3, . . ., we get
P (X = x) = p × P (Y = r − 1)
= p × FBinomial(x−1,p) (r − 1) − FBinomial(x−1,p) (r − 2) .
(2.41)
•
Exercise 2.68 — Relating the d.f. of the negative binomial and binomial r.v.
Let X ∼ BinomialN(r, p), Y ∼ Binomial(x, p) e Z = x − Y ∼ Binomial(x, 1 − p). Prove
that, for x = r, r + 1, r + 2, . . . and r = 1, 2, 3, . . ., we have
FBinomialN (r,p) (x) = P (X ≤ x)
= P (Y ≥ r)
= 1 − FBinomial(x,p) (r − 1)
= P (Z ≤ x − r)
= FBinomial(x,1−p) (x − r).
(2.42)
•
73
2.4.2
Absolutely continuous r.v. and random vectors
• Uniform distribution on the interval [a, b]
Notation
X ∼ Uniform(a, b)
Parameters
a = minimum value (a ∈ IR)
b = maximum value (b ∈ IR, a < b)
Range
[a, b]
P.d.f.
fX (x) =
1
b−a ,
a≤x≤b
Let X be an absolutely continuous r.v. with d.f. FX (x). Then Y = FX (X) ∼
Uniform(0, 1).
• Beta distribution
In probability theory and statistics, the beta distribution is a family of continuous
probability distributions defined on the interval [0, 1] parameterized by two positive
shape parameters, typically denoted by α and β. In Bayesian statistics, it can be
seen as the posterior distribution of the parameter p of a binomial distribution,
if the prior distribution of p was uniform. It is also used in information theory,
particularly for the information theoretic performance analysis for a communication
system.
Notation
X ∼ Beta(α, β)
Parameters
α (α ∈ IR+ )
β (β ∈ IR+ )
Range
[0, 1]
P.d.f.
fX (x) =
1
B(α,β)
xα−1 (1 − x)β−1 , 0 ≤ x ≤ 1
where
Z
B(α, β) =
1
xα−1 (1 − x)β−1 dx
(2.43)
0
represents the beta function. Note that
B(α, β) =
Γ(α)Γ(β)
,
Γ(α + β)
(2.44)
74
where
Z
Γ(α) =
+∞
y α−1 e−y dy
(2.45)
0
is the Euler’s gamma function.
The uniform distribution on [0, 1] is a particular case of the beta distribution —
α = β = 1. Moreover, the beta distribution can be generalized to the interval [a, b]:
fY (y) =
1
(y − a)α−1 (b − y)β−1
, a ≤ y ≤ b.
B(α, β)
(b − a)α+β−1
(2.46)
The p.d.f. of this distribution can take various forms on account of the “shape”
parameters a and b, as illustrated by the following graph and table:
Parameters
α, β > 1
α < 1, β > 1
(α − 1)(β − 1) ≤ 0
α=β
α<β
α>β
Shape of the beta p.d.f.
α−1
Unique mode at x = α+β−2
α−1
Unique anti-mode at x = α+β−2
(U − shape)
J − shape
Symmetric around 1/2 (e.g. constant ou parabolic)
Positively assymmetric
Negatively assymmetric
Exercise 2.69 — Relating the Beta and Binomial distributions
(a) Prove that the d.f. of the r.v. X ∼ Beta(α, β) can be written in terms of the
d.f. of Binomial r.v. when α and β are integer-valued:
FBeta(α,β) (x) = 1 − FBinomial(α+β−1,x) (α − 1).
75
(2.47)
(b) Prove that the p.d.f. of the r.v. X ∼ Beta(α, β) can be rewritten in terms of the
p.f. of the r.v. Y ∼ Binomial(α + β − 2, x), when α and β are integer-valued:
fBeta(α,β) (x) = (α + β − 1) × P (Y = α − 1)
= (α + β − 1) × FBinomial(α+β−2,x) (α − 1)
− FBinomial(α+β−2,x) (α − 2) .
(2.48)
•
• Normal distribution
The normal distribution or Gaussian distribution is a continuous probability
distribution that describes data that cluster around a mean or average. The graph of
the associated probability density function is bell-shaped, with a peak at the mean,
and is known as the Gaussian function or bell curve. The Gaussian distribution
is one of many things named after Carl Friedrich Gauss, who used it to analyze
astronomical data, and determined the formula for its probability density function.
However, Gauss was not the first to study this distribution or the formula for its
density function that had been done earlier by Abraham de Moivre.
Notation
X ∼ Normal(µ, σ 2 )
Parameters
µ (µ ∈ IR)
σ 2 (σ 2 ∈ IR+ )
Range
IR
P.d.f.
fX (x) =
√ 1 e−
2πσ
(x−µ)2
2σ 2
, −∞ < x < +∞
The normal distribution can be used to describe, at least approximately, any variable
that tends to cluster around the mean. For example, the heights of adult males in
the United States are roughly normally distributed, with a mean of about 1.8 m.
Most men have a height close to the mean, though a small number of outliers have
a height significantly above or below the mean. A histogram of male heights will
appear similar to a bell curve, with the correspondence becoming closer if more data
are used. (http://en.wikipedia.org/wiki/Normal distribution).
Standard normal distribution — Let X ∼ Normal(µ, σ 2 ). Then the r.v. Z =
X−E(X)
√
= X−µ
is said to have a standard normal distribution, i.e. Z ∼ Normal(0, 1).
σ
V (X)
Moreover, Z has d.f. given by
Z z
t2
1
√ e− 2 dt = Φ(z),
FZ (z) = P (Z ≤ z) =
2π
−∞
76
(2.49)
and
FX (x) = P (X ≤ x)
X −µ
x−µ
= P Z=
≤
σ
σ
x−µ
.
= Φ
σ
(2.50)
• Exponential distribution
The exponential distributions are a class of continuous probability distributions.
They tend to be used to describe the times between events in a Poisson process,
i.e. a process in which events occur continuously and independently at a constant
average rate (http://en.wikipedia.org/wiki/Exponential distribution).
Notation
X ∼ Exponential(λ)
Parameter
λ = inverse of the scale parameter (λ ∈ IR+ )
Range
IR0+ = [0, +∞)
P.d.f.
fX (x) = λ e−λx , x ≥ 0
Consider X ∼ Exponencial(λ). Then
P (X > t + x|X > t) = P (X > x), ∀t, x ∈ IR0+ .
(2.51)
Equivalently,
(X − t|X > t) ∼ Exponencial(λ), ∀t ∈ IR0+ .
(2.52)
This property is referred as to lack of memory: no matter how old your equipment
is, its remaining life has same distribution as a new one.
The exponential (resp. geometric) distribution is the only absolutely continuous
(resp. discrete) r.v. satisfying this property.
Poisson process — We can relate exponential and Poisson r.v. as follows. Let:
– X be the time between two consecutive events;
– Nx the number of times the event has occurred in the interval (0, x].
Then
Nx ∼ Poisson(λ × x) ⇔ X ∼ Exponential(λ)
(2.53)
and the collection of r.v. {Nx : x > 0} is said to be a Poisson process with rate λ.
77
• Gamma distribution
The gamma distribution is frequently a probability model for waiting
times; for instance, in life testing, the waiting time until death is a
random variable that is frequently modeled with a gamma distribution
(http://en.wikipedia.org/wiki/Gamma distribution).
Notation
X ∼ Gamma(α, β)
Parameters
α = shape parameter (α ∈ IR+ )
β = inverse of the scale parameter (β ∈ IR+ )
Range
IR0+ = [0, +∞)
P.d.f.
fX (x) =
βα
Γ(α)
xα−1 e−βx , x ≥ 0
Special cases
– Exponential — α = 1 which has the lack of memory property as the geometric
distribution in the discrete case;
– Erlang — α ∈ IN ;5
– Chi-square with n degrees of freedom — α = n/2, β = 1/2.
This distribution has a shape parameter α, therefore it comes as no surprise the
sheer variety of forms of the gamma p.d.f. in the following graph.
Parameters
α<1
α=1
α>1
Shape of the gamma p.d.f.
Unique supremum at x = 0
Unique mode at x = 0
Unique mode at x = α−1
β and positively assymmetric
The gamma distribution stand in the same relation to exponential as negative
binomial to geometric: sums of i.i.d exponential r.v. have gamma distribution.
χ2 distributions result from sums of squares of independent standard normal r.v.
5
The Erlang distribution was developed by Agner Krarup Erlang (1878–1929) to examine the number
of telephone calls which might be made at the same time to the operators of the switching stations. This
work on telephone traffic engineering has been expanded to consider waiting times in queueing systems
in general. The distribution is now used in the fields of stochastic processes and of biomathematics
(http://en.wikipedia.org/wiki/Erlang distribution)
78
It is possible to relate the d.f. of X ∼ Erlang(n, β) with the d.f. of a Poisson r.v.:
FErlang(n,β) (x) =
∞
X
e−βx (βx)i /i!
i=n
= 1 − FP oisson(βx) (n − 1), x > 0, n ∈ IN.
(2.54)
• d−dimensional uniform distribution
Notation
X ∼ Uniform([0, 1d ])
Range
[0, 1]d
P.d.f.
fX (x) = 1, x ∈ [0, 1]d
• Bivariate STANDARD normal distribution
"
0
0
# "
1 ρ
ρ 1
#!
Notation
X ∼ Normal
Parameter
Range
ρ = correlation between X1 and X2 (−1 ≤ ρ ≤ 1)
IR2
x2 −2ρx x +x2
fX (x) = f(X1 ,X2 ) (x1 , x2 ) = √1 2 exp − 12 1 1−ρ1 22 2 , x ∈ IR2
P.d.f.
,
2π
1−ρ
The graphical representation of the joint density of a random vector with a bivariate
standard normal distribution follows — it depends on the parameter ρ.
79
Case
ρ=0
Graph and contour plot of the joint p.d.f.
of a bivariate STANDARD normal
Circumferences centered in (0, 0)
3
2
1
0.15
0.1
0.05
0
0
2
0 x2
-2
0
x1
ρ<0
2
-1
-2
-2
-3
-3
-2
-1
0
1
2
3
Ellipses centered in (0, 0) and asymmetric in relation to the axes,
suggesting that X2 decreases when X1 increases
3
2
1
0.3
0.2
0.1
0
0
2
-1
0 x2
-2
0
x1
ρ>0
2
-2
-2
-3
-3
-2
-1
0
1
2
3
Ellipses centered in (0, 0) and asymmetric in relation to the axes,
suggesting that X2 increases when X1 increases
3
2
1
0.3
0.2
0.1
0
0
2
-1
0 x2
-2
0
x1
2
-2
-2
-3
-3
-2
-1
0
1
2
3
Both components of X = (X1 , X2 ) have standard normal marginal densities and are
independent iff ρ = 0.
80
2.5
Transformation theory
2.5.1
Transformations of r.v., general case
Motivation 2.70 — Transformations of r.v., general case (Karr, 1993, p. 60)
Let:
• X be a r.v. with d.f. FX ;
• Y = g(X) be a transformation of X under g, where g : IR → IR is a Borel measurable
function.
Then we know that Y = g(X) is also a r.v. But this is manifestly not enough: we wish
to know
• how the d.f. of Y relates to that of X?
This question admits an obvious answer when g is invertible and in a few other cases
described below.
•
Proposition 2.71 — D.f. of a transformation of a r.v., general case (Rohatgi,
1976, p. 68; Murteira, 1979, p. 121)
Let:
• X be a r.v. with d.f. FX ;
• Y = g(X) be a transformation of X under g, where g : IR → IR is a Borel measurable
function;
• g −1 ((−∞, y]) = {x ∈ IR : g(x) ≤ y} be the inverse image of the Borel set (−∞, y]
under g.
Then
FY (y) = P ({Y ≤ y})
= P ({X ∈ g −1 ((−∞, y])}).
(2.55)
•
Exercise 2.72 — D.f. of a transformation of a r.v., general case
Prove Proposition 2.71 (Rohatgi, 1976, p. 68).
Note that if g is a Borel measurable function then
g −1 (B) ∈ B(IR), ∀B = (−∞, y] ∈ B(IR).
81
(2.56)
Thus, we are able to write
P ({Y ∈ B}) = P ({g(X) ∈ B}) = P ({X ∈ g −1 (B)}).
(2.57)
•
Remark 2.73 — D.f. of a transformation of a r.v., general case
Proposition 2.71 relates the d.f. of Y to that of X.
The inverse image g −1 ((−∞, y]) is a Borel set and tends to be a “reasonable” set —
a real interval or a union of real intervals.
•
Exercise 2.74 — D.f. of a transformation of a r.v., general case (Karr, 1993, p.
70, Exercise 2.20(a))
Let X be a r.v. and Y = X 2 . Prove that
√
√
(2.58)
FY (y) = FX ( y) − FX [−( y)− ],
for y ≥ 0.
•
Exercise 2.75 — D.f. of a transformation of a r.v., general case (Rohatgi, 1976,
p. 68)
Let X be a r.v. with d.f. FX . Derive the d.f. of the following r.v.:
(a) |X|
(b) aX + b
(c) eX .
•
Exercise 2.76 — D.f. of a transformation of a r.v., absolutely continuous case
The electrical resistance6 (X) of an object and its electrical conductance7 (Y ) are related
as follows: Y = X −1 .
Assuming that X ∼ Uniform(900 ohm, 1100 ohm):
(a) Identify the range of values of the r.v. Y .
(b) Derive the survival function of Y , P (Y > y), and calculate P (Y > 10−3 mho).
6
•
The electrical resistance of an object is a measure of its opposition to the
passage of a steady electric current.
The SI unit of electrical resistance is the ohm
(http://en.wikipedia.org/wiki/Electrical resistance).
7
Electrical conductance is a measure of how easily electricity flows along a certain path through an
electrical element. The SI derived unit of conductance is the siemens (also called the mho, because it is
the reciprocal of electrical resistance, measured in ohms). Oliver Heaviside coined the term in September
1885 (http://en.wikipedia.org/wiki/Electrical conductance).
82
Exercise 2.77 — D.f. of a transformation of a r.v., absolutely continuous case
Let X ∼ Uniform(0, 2π) and Y = sin X. Prove that


y < −1
 0,
arcsin y
1
(2.59)
FY (y) =
+ π , −1 ≤ y ≤ 1
2


1,
y > 1.
•
2.5.2
Transformations of discrete r.v.
Proposition 2.78 — P.f. of a one-to-one transformation of a discrete r.v.
(Rohatgi, 1976, p. 69)
Let:
• X be a discrete r.v. with p.f. P ({X = x});
• RX be a countable set such that P ({X ∈ RX }) = 1 and P ({X = x}) > 0,
∀x ∈ RX ;
• Y = g(X) be a transformation of X under g, where g : IR → IR is a one-to-one
Borel measurable function that transforms RX onto some set RY = g(RX ).
Then the inverse map, g −1 , is a single-valued function of y and
(
P ({X = g −1 (y)}), y ∈ RY
P ({Y = y}) =
0,
otherwise.
(2.60)
•
Exercise 2.79 — P.f. of a one-to-one transformation of a discrete r.v. (Rohatgi,
1976, p. 69)
Let X ∼ Poisson(λ). Obtain the p.f. of Y = X 2 + 3.
•
Exercise 2.80 — P.f. of a one-to-one transformation of a discrete r.v.
Let X ∼ Binomial(n, p) and Y = n − X. Prove that:
• Y ∼ Binomial(n, 1 − p);
• FY (y) = 1 − FX (n − y − 1), y = 0, 1, . . . , n.
•
Remark 2.81 — P.f. of a transformation of a discrete r.v. (Rohatgi, 1976, p. 69)
Actually the restriction of a single-valued inverse on g is not necessary. If g has a finite (or
even a countable) number of inverses for each y, from the countable additivity property
of probability functions we can obtain the p.f. of the r.v. Y = g(X).
•
83
Proposition 2.82 — P.f. of a transformation of a discrete r.v. (Murteira, 1979,
p. 122)
Let:
• X be a discrete r.v. with p.f. P ({X = x});
• RX be a countable set such that P ({X ∈ RX }) = 1 and P ({X = x}) > 0,
∀x ∈ RX ;
• Y = g(X) be a transformation of X under g, where g : IR → IR is a Borel measurable
function that transforms RX onto some set RY = g(RX );
• Ay = {x ∈ RX : g(x) = y} be a non empty set, for y ∈ RY .
Then
P ({Y = y}) = P ({X ∈ Ay })
X
P ({X = x}),
=
(2.61)
x∈Ay
for y ∈ RY .
•
Exercise 2.83 — P.f. of a transformation of a discrete r.v. (Rohatgi, 1976, pp.
69–70)
Let X be a discrete r.v. with p.f.
 1
, x = −2


5


1

, x = −1

6


 1, x = 0
5
(2.62)
P ({X = x}) =
1

,
x
=
1

15


11


, x=2


 30
0, otherwise
Derive the p.f. of Y = X 2 .
•
84
2.5.3
Transformations of absolutely continuous r.v.
Proposition 2.84 — D.f. of a strictly monotonic transformation of an
absolutely continuous r.v. (Karr, 1993, pp. 60 and 68)
Let:
• X be an absolutely continuous r.v. with d.f. FX and p.d.f. fX ;
• RX be the range of the r.v. X, i.e. RX = {x ∈ IR : fX (x) > 0};
• Y = g(X) be a transformation of X under g, where g : IR → IR is a continuous,
strictly increasing, Borel measurable function that transforms RX onto some set
RY = g(RX );
• g −1 be the pointwise inverse of g.
Then
FY (y) = FX [g −1 (y)],
(2.63)
for y ∈ RY . Similarly, if
• g is a continuous, strictly decreasing, Borel measurable function
then
FY (y) = 1 − FX [g −1 (y)],
(2.64)
for y ∈ RY .
•
Exercise 2.85 — D.f. of a strictly monotonic transformation of an absolutely
continuous r.v.
Prove Proposition 2.84 (Karr, 1993, p. 60).
•
Exercise 2.86 — D.f. of a strictly monotonic transformation of an absolutely
continuous r.v.
Let X ∼ Normal(0, 1). Derive the d.f. of
(a) Y = eX
(b) Y = µ + σX, where µ ∈ IR and σ ∈ IR+
•
(Karr, 1993, p. 60).
85
Remark 2.87 — Transformations of absolutely continuous and discrete r.v.
(Karr, 1993, p. 61)
in general, Y = g(X) need not be absolutely continuous even when X is, as shown in the
next exercise, while if X is a discrete r.v. then so is Y = g(X) regardless of the Borel
measurable function g.
•
Exercise 2.88 — A mixed r.v. as a transformation of an absolutely continuous
r.v.
Let X ∼ Uniform(−1, 1). Prove that Y = X + = max{0, X} is a mixed r.v. whose d.f. is
given by

0,
y<0



 1,
y=0
2
(2.65)
FY (y) =
y
1

+
,
0
<
y
≤
1


 2 2
1,
y>1
•
(Rohatgi, 1976, p. 70).
Exercise 2.88 shows that we need some conditions on g to ensure that Y = g(X) is
also an absolutely continuous r.v. This will be the case when g is a continuous monotonic
function.
Theorem 2.89 — P.d.f. of a strictly monotonic transformation of an absolutely
continuous r.v. (Rohatgi, 1976, p. 70; Karr, 1993, p. 61)
Suppose that:
• X is an absolutely continuous r.v. with p.d.f. fX ;
• there is an open subset RX ⊂ IR such that P ({X ∈ RX }) = 1;
• Y = g(X) is a transformation of X under g, where g : IR → IR is a continuously
differentiable, Borel measurable function such that either dg(x)
> 0, ∀x ∈ RX , or
dx
dg(x)
< 0, ∀x ∈ RX ;8
dx
• g transforms RX onto some set RY = g(RX );
• g −1 represents the pointwise inverse of g.
8
This implies that
dg(x)
dx
6= 0, ∀x ∈ RX .
86
Then Y = g(X) is an absolutely continuous r.v. with p.d.f. given by
dg −1 (y)
,
fY (y) = fX [g (y)] ×
dy
−1
(2.66)
for y ∈ RY .
•
Exercise 2.90 — P.d.f. of a strictly monotonic transformation of an absolutely
continuous r.v.
Prove Theorem 2.89 by considering the case dg(x)
> 0, ∀x ∈ RX , applying Proposition 2.84
dx
to derive the d.f. of Y = g(X), and differentiating it to obtain the p.d.f. of Y (Rohatgi,
1976, p. 70).
•
Remark 2.91 — P.d.f. of a strictly monotonic transformation of an absolutely
continuous r.v. (Rohatgi, 1976, p. 71)
The key to computation of the induced d.f. of Y = g(X) from the d.f. of X is P ({Y ≤
y}) = P ({X ∈ g −1 ((−∞, y])}). If the conditions of Theorem 2.89 are satisfied, we are
able to identify the set {X ∈ g −1 ((−∞, y])} as {X ≤ g −1 (y)} or {X ≥ g −1 (y)}, according
to whether g in strictly increasing or strictly decreasing.
•
Exercise 2.92 — P.d.f. of a strictly monotonic transformation of an absolutely
continuous r.v.
Let X ∼ Normal(0, 1). Identify the p.d.f. and the distribution of
(a) Y = eX
(b) Y = µ + σX, where µ ∈ IR and σ ∈ IR+
•
(Karr, 1993, p. 61).
Corollary 2.93 — P.d.f. of a strictly monotonic transformation of an absolutely
continuous r.v. (Rohatgi, 1976, p. 71)
Under the conditions of Theorem 2.89, and by noting that
dg −1 (y)
=
dy
1
dg(x)
dx x=g −1 (y)
,
(2.67)
we conclude that the p.d.f. of Y = g(X) can be rewritten as follows:
fY (y) =
fX (x)
dg(x)
dx
,
(2.68)
x=g −1 (y)
∀y ∈ RY .
•
87
Remark 2.94 — P.d.f. of a non monotonic transformation of an absolutely
continuous r.v. (Rohatgi, 1976, p. 71)
In practice Theorem 2.89 is quite useful, but whenever its conditions are violated we
should return to P ({Y ≤ y}) = P ({X ∈ g −1 ((−∞, y])}) to obtain the FY (y) and then
differentiate this d.f. to derive the p.d.f. of the transformation Y . This is the case in the
next two exercises.
•
Exercise 2.95 — P.d.f. of a non monotonic
continuous r.v.
Let X ∼ Normal(0, 1) and Y = g(X) = X 2 . Prove
√
√
FY (y) = FX ( y) − FX (− y), y > 0
dFY (y)
fY (y) =
dy
(
√
√ 1
√
×
f
(
y)
+
f
(−
y) ,
X
X
2 y
=
0,
transformation of an absolutely
that Y ∼ χ2(1) by noting that
(2.69)
y≥0
y<0
(2.70)
•
(Rohatgi, 1976, p. 72).
Exercise 2.96 — P.d.f. of a non monotonic transformation of an absolutely
continuous r.v.
Let X be an absolutely continuous r.v. with p.d.f.
(
2x
, 0<x<π
π2
fX (x) =
(2.71)
0, otherwise
Prove that Y = sin X has p.d.f. given by
(
√2 , 0 < y < 1
π 1−y 2
fY (y) =
0,
otherwise
(2.72)
•
(Rohatgi, 1976, p. 73).
Motivation 2.97 — P.d.f. of a sum of monotonic restrictions of a function g of
an absolutely continuous r.v. (Rohatgi, 1976, pp. 73–74)
in the two last exercises the function y = g(x) can be written as the sum of two monotonic
restrictions of g in two disjoint intervals. Therefore we can apply Theorem 2.89 to each
of these monotonic summands.
In fact, these two exercises are special cases of the following theorem.
•
88
Theorem 2.98 — P.d.f. of a finite sum of monotonic restrictions of a function
g of an absolutely continuous r.v. (Rohatgi, 1976, pp. 73–74)
Let:
• X be an absolutely continuous r.v. with p.d.f. fX ;
• Y = g(X) be a transformation of X under g, where g : IR → IR is a Borel measurable
function that transforms RX onto some set RY = g(RX ).
Moreover, suppose that:
• g(x) is differentiable for all x ∈ RX ;
•
dg(x)
dx
is continuous and nonzero at all points of RX but a finite number of x.
Then, for every real number y ∈ RY ,
(a) there exists a positive integer n
g1−1 (y), . . . , gn−1 (y) such that
g(x)|x=g−1 (y) = y
k
and
dg(x)
dx
=
n(y) and real numbers (inverses)
6= 0, k = 1, . . . , n(y),
(2.73)
x=gk−1 (y)
or
(b) there does not exist any x such that g(x) = y and
n = n(y) = 0.
dg(x)
dx
6= 0, in which case we write
In addition, Y = g(X) is an absolutely continuous r.v. with p.d.f. given by
( Pn(y)
dgk−1 (y)
−1
, n = n(y) > 0
k=1 fX [gk (y)] ×
dy
fY (y) =
0,
n = n(y) = 0,
for y ∈ RY .
(2.74)
•
Exercise 2.99 — P.d.f. of a finite sum of monotonic restrictions of a function
g of an absolutely continuous r.v.
Let X ∼ Uniform(−1, 1). Use Theorem 2.98 to prove that Y = |X| ∼ Uniform(0, 1)
(Rohatgi, 1976, p. 74).
•
89
Exercise 2.100 — P.d.f. of a finite sum of monotonic restrictions of a function
g of an absolutely continuous r.v.
Let X ∼ Uniform(0, 2π) and Y = sin X. Use Theorem 2.98 to prove that
(
√1 , −1 < y < 1
π
1−y 2
fY (y) =
(2.75)
0,
otherwise.
•
Motivation 2.101 — P.d.f. of a countable sum of monotonic restrictions of a
function g of an absolutely continuous r.v.
The formula P ({Y ≤ y}) = P ({X ∈ g −1 ((−∞, y])}) and the countable additivity of
probability functions allows us to compute the p.d.f. of Y = g(X) in some instance even
if g has a countable number of inverses.
•
Theorem 2.102 — P.d.f. of a countable sum of monotonic restrictions of a
function g of an absolutely continuous r.v. (Rohatgi, 1976, pp. 74–75)
Let g be a Borel measurable function that maps RX onto some set RY = g(RX ). Suppose
that RX can be represented as a countable union of disjoint sets Ak , k = 1, 2, . . . Then
Y = g(X) is an absolutely continuous r.v. with d.f. given by
FY (y) = P ({Y ≤ y})
= P ({X ∈ g −1 ((−∞, y])})
(
)!
+∞
[
= P
X∈
g −1 ((−∞, y]) ∩ Ak
k=1
=
+∞
X
P
X ∈ g −1 ((−∞, y]) ∩ Ak
(2.76)
k=1
for y ∈ RY .
If the conditions of Theorem 2.89 are satisfied by the restriction of g to each Ak , gk ,
we may obtain the p.d.f. of Y = g(X) on differentiating the d.f. of Y .9 In this case
fY (y) =
+∞
X
k=1
fX [gk−1 (y)] ×
dgk−1 (y)
dy
(2.77)
for y ∈ RY .
•
9
We remind the reader that term-by-term differentiation is permissible if the differentiated series is
uniformly convergent.
90
Exercise 2.103 — P.d.f. of a countable sum of monotonic restrictions of a
function g of an absolutely continuous r.v.
Let X ∼ Exponential(λ) and Y = sin X. Prove that
e−λπ+λ arcsin y − e−λ arcsin y
FY (y) = 1 +
,0<y<1
1 − e−2πλ

λ arcsin y
λe−λπ√
−λπ−λ arcsin y
×
e
+
e
, −1 ≤ y < 0


 (1−e−2λπ )× 1−y2 λ √
fY (y) =
× e−λ arcsin y + e−λπ+λ arcsin y , 0 ≤ y < 1
−2λπ )×
2
(1−e
1−y



0,
otherwise
(2.79)
•
(Rohatgi, 1976, p. 75).
2.5.4
(2.78)
Transformations of random vectors, general case
What follows is the analogue of Proposition 2.71 in a multidimensional setting.
Proposition 2.104 — D.f. of a transformation of a random vector, general case
Let:
• X = (X1 , . . . , Xd ) be a random vector with joint d.f. FX ;
• Y = (Y1 , . . . , Ym ) = g(X) = (g1 (X1 , . . . , Xd ), . . . , gm (X1 , . . . , Xd )) be a
transformation of X under g, where g : IRd → IRm is a Borel measurable function;
Q
• g −1 ( m
i=1 (−∞, yi ]) = {x = (x1 , . . . , xd ) ∈ IR : g1 (x1 , . . . , xd ) ≤ y1 , . . .,
Q
gm (x1 , . . . , xd ) ≤ ym } be the inverse image of the Borel set m
i=1 (−∞, yi ] under
10
g.
Then
FY (y) = P ({Y1 ≤ y1 , . . . , Ym ≤ ym })
m
Y
= P ({X ∈ g −1 ( (−∞, yi ])}).
(2.80)
i=1
•
Exercise 2.105 — D.f. of a transformation of a random vector, general case
Let X = (X1 , . . . , Xd ) be an absolutely continuous random vector such that
indep
Xi ∼ Exponential(λi ), i = 1, . . . , d.
P
Prove that Y = mini=1,...,d Xi ∼ Exponential( di=1 λi ).
•
Let us remind the reader that since g is a Borel measurable function we have g −1 (B) ∈ B(IRd ), ∀B ∈
B(IRm ).
10
91
2.5.5
Transformations of discrete random vectors
Theorem 2.106 — Joint p.f. of a one-to-one transformation of a discrete
random vector (Rohatgi, 1976, p. 131)
Let:
• X = (X1 , . . . , Xd ) be a discrete random vector with joint p.f. P ({X = x});
• RX be a countable set of points such that P (X ∈ RX ) = 1 and P ({X = x) > 0,
∀x ∈ IRX ;
• Y = (Y1 , . . . , Yd ) = g(X) = (g1 (X1 , . . . , Xd ), . . . , gd (X1 , . . . , Xd )) be a
transformation of X under g, where g : IRd → IRd is a one-to-one Borel measurable
function that maps RX onto some set RY ⊂ IRd ;
• g −1 be the inverse mapping such that g −1 (y) = (g1−1 (y), . . . , gd−1 (y)).
Then the joint p.f. of Y = (Y1 , . . . , Yd ) is given by
P ({Y = y}) = P ({Y1 = y1 , . . . , Yd = yd })
= P ({X1 = g1−1 (y), . . . , Xd = gd−1 (y)}),
(2.81)
for y = (y1 , . . . , yd ) ∈ RY .
•
Remark 2.107 — Joint p.f. of a one-to-one transformation of a discrete random
vector (Rohatgi, 1976, pp. 131–132)
The marginal p.f. of any Yj (resp. the joint p.f. of any subcollection of Y1 , . . . , Yd ,
say (Yj )j∈I⊂{1,...,d} ) is easily computed by summing on the remaining yi , i 6= j (resp.
(Yi )i6∈I ).
•
Theorem 2.108 — Joint p.f. of a transformation of a discrete random vector
Let:
• X = (X1 , . . . , Xd ) be a discrete random vector with range RX ⊂ IRd ;
• Y = (Y1 , . . . , Ym ) = g(X) = (g1 (X1 , . . . , Xd ), . . . , gm (X1 , . . . , Xd )) be a
transformation of X under g, where g : IRd → IRm is a Borel measurable function
that maps RX onto some set RY ⊂ IRm ;
• Ay1 ,...,ym = {x = (x1 , . . . , xd ) ∈ RX : g1 (x1 , . . . , xd ) = y1 , . . . , gm (x1 , . . . , xd ) = ym }.
Then the joint p.f. of Y = (Y1 , . . . , Ym ) is given by
P ({Y = y}) = P ({Y1 = y1 , . . . , Ym = ym })
X
=
P ({X1 = x1 , . . . , Xd = xd }),
(2.82)
x=(x1 ,...,xd )∈Ay1 ,...,ym
for y = (y1 , . . . , yd ) ∈ RY .
•
92
Exercise 2.109 — Joint p.f. of a transformation of a discrete random vector
Let X = (X1 , X2 ) be a discrete random vector with joint p.f. P (X = x, Y = y) given in
the following table:
X1
−1
0
1
-2
X2
0
1
6
1
12
1
6
1
6
1
12
1
6
Derive the joint p.f. of Y1 = |X1 | and Y2 = X22 .
2
1
12
0
1
12
•
Theorem 2.110 — P.f. of the sum, difference, product and division of two
discrete r.v.
Let:
• (X, Y ) be a discrete bidimensional random vector with joint p.f. P (X = x, Y = y);
• Z =X +Y
• U =X −Y
• V =XY
• W = X/Y , provided that P ({Y = 0}) = 0.
Then
P (Z = z) = P (X + Y = z)
X
=
P (X = x, X + Y = z)
x
=
X
P (X = x, Y = z − x)
x
=
X
P (X + Y = z, Y = y)
y
=
X
P (X = z − y, Y = y)
y
P (U = u) = P (X − Y = u)
X
=
P (X = x, X − Y = u)
x
=
X
P (X = x, Y = x − u)
x
93
(2.83)
=
X
P (X − Y = u, Y = y)
y
=
X
P (X = u + y, Y = y)
(2.84)
y
P (V = v) = P (X Y = v)
X
=
P (X = x, XY = v)
x
=
X
=
X
P (X = x, Y = v/x)
x
P (XY = v, Y = y)
y
=
X
P (X = v/y, Y = y)
(2.85)
y
P (W = w) = P (X/Y = w)
X
=
P (X = x, X/Y = w)
x
=
X
=
X
=
X
P (X = x, Y = x/w)
x
P (X/Y = w, Y = y)
y
P (X = wy, Y = y).
(2.86)
y
•
Exercise 2.111 — P.f. of the difference of two discrete r.v.
Let (X, Y ) be a discrete random vector with joint p.f. P (X = x, Y = y) given in the
following table:
X
1
2
3
1
1
12
2
12
1
12
Y
2
3
1
12
2
12
0
0
1
12
4
12
(a) Prove that X and Y are identically distributed but are not independent.
(b) Obtain the p.f. of U = X − Y
(c) Prove that U = X − Y is not a symmetric r.v., that is U and −U are not identically
distributed.
•
94
Corollary 2.112 — P.f. of the sum, difference, product and division of two
independent discrete r.v.
Let:
• X and Y be two independent discrete r.v. with joint p.f. P (X = x, Y = y) =
P (X = x) × P (Y = y), ∀x, y
• Z =X +Y
• U =X −Y
• V =XY
• W = X/Y , provided that P ({Y = 0}) = 0.
Then
P (Z = z) = P (X + Y = z)
X
=
P (X = x) × P (Y = z − x)
x
=
X
P (X = z − y) × P (Y = y)
(2.87)
y
P (U = u) = P (X − Y = u)
X
P (X = x) × P (Y = x − u)
=
x
=
X
P (X = u + y) × P (Y = y)
(2.88)
y
P (V = v) = P (X Y = v)
X
P (X = x) × P (Y = v/x)
=
x
=
X
P (X = v/y) × P (Y = y)
(2.89)
y
P (W = w) = P (X/Y = w)
X
=
P (X = x) × P (Y = x/w)
x
=
X
P (X = wy) × P (Y = y).
(2.90)
y
•
95
Exercise 2.113 — P.f. of the sum of two independent r.v. with three well
known discrete distributions
Let X and Y be two independent discrete r.v. Prove that
(a) X ∼ Binomial(nX , p) ⊥⊥ Y ∼ Binomial(nY , p) ⇒ (X + Y ) ∼ Binomial(nX + nY , p)
(b) X ∼ NegativeBinomial(nX , p) ⊥⊥ Y ∼ NegativeBinomial(nX , p) ⇒ (X + Y ) ∼
NegativeBinomial(nX + nY , p)
(c) X ∼ Poisson(λX ) ⊥⊥ Y ∼ Poisson(λY ) ⇒ (X + Y ) ∼ Poisson(λX + λY ),
i.e. the families of Poisson, Binomial and Negative Binomial distributions are closed under
summation of independent members.
•
Exercise 2.114 — P.f. of the difference of two independent Poisson r.v.
Let X ∼ Poisson(λX ) ⊥⊥ Y ∼ Poisson(λY ). Then (X − Y ) has p.f. given by
+∞
X
P (X = u + y) × P (Y = y)
P (X − Y = u) =
y=0
= e−(λX +λY )
+∞
X
y=max{0,−u}
y
λu+y
X λY
, u = . . . , −1, 0, 1, . . .
(u + y)! y!
(2.91)
•
Remark 2.115 —
Skellam
distribution
(http://en.wikipedia.org/wiki/
Skellam distribution)
The Skellam distribution is the discrete probability distribution of the difference of two
correlated or uncorrelated r.v. X and Y having Poisson distributions with parameters λX
and λY . It is useful in describing the statistics of the difference of two images with simple
photon noise, as well as describing the point spread distribution in certain sports where
all scored points are equal, such as baseball, hockey and soccer.
When λX = λY = λ and u is also large, and of order of the square root of 2λ,
u2
e− 2×2λ
P (X − Y = u) ' √
,
2π × 2λ
(2.92)
the p.d.f. of a Normal distribution with parameters µ = 0 and σ 2 = 2λ.
Please note that the expression of the p.f. of the Skellam distribution that can be
found in http://en.wikipedia.org/wiki/Skellam distribution is not correct.
•
96
2.5.6
Transformations of absolutely continuous random vectors
Motivation 2.116 — P.d.f. of a transformation of an absolutely continuous
random vector (Karr, 1993, p. 62)
Recall that a random vector X = (X1 , . . . , Xd ) is absolutely continuous if there is a
function fX on Rd satisfying
FX (x) = FX1 ,...,Xd (x1 , . . . , xd )
Z xd
Z x1
fX1 ,...,Xd (s1 , . . . , sd ) dsd . . . ds1 .
...
=
(2.93)
−∞
−∞
Computing the density of Y = g(X) requires that g be invertible, except for the special
•
case that X1 , . . . , Xd are independent (and then only for particular choices of g).
Theorem 2.117 — P.d.f. of a one-to-one transformation of an absolutely
continuous random vector (Rohatgi, 1976, p. 135; Karr, 1993, p. 62)
Let:
• X = (X1 , . . . , Xd ) be an absolutely continuous random vector with joint p.d.f. fX (x);
• RX be an open set of IRd such that P (X ∈ RX ) = 1;
• Y = (Y1 , . . . , Yd ) = g(X) = (g1 (X1 , . . . , Xd ), . . . , gd (X1 , . . . , Xd )) be a
transformation of X under g, where g : IRd → IRd is a one-to-one Borel measurable
function that maps RX onto some set RY ⊂ IRd ;
• g −1 (y) = (g1−1 (y), . . . , gd−1 (y)) be the inverse mapping defined over the range RY of
the transformation.
Assume that:
• both g and its inverse g −1 are continuous;
• the partial derivatives,
∂gi−1 (y)
,
∂yj
1 ≤ i, j ≤ d, exist and are continuous;
• the Jacobian of the inverse transformation g −1 (i.e. the determinant of the matrix
of partial derivatives
∂g1−1 (y)
∂y1
J(y) =
..
.
∂gd−1 (y)
∂y1
∂gi−1 (y)
)
∂yj
···
···
···
is such that
∂g1−1 (y)
∂yd
..
.
6= 0,
∂gd−1 (y)
∂yd
for y = (y1 , . . . , yd ) ∈ RY .
97
(2.94)
Then the random vector Y = (Y1 , . . . , Yd ) is absolutely continuous and its joint p.d.f. is
given by
(2.95)
fY (y) = fX g −1 (y) × |J(y)|,
for y = (y1 , . . . , yd ) ∈ RY .
•
Exercise 2.118 — P.d.f. of a one-to-one transformation of an absolutely
continuous random vector
Prove Theorem 2.117 (Rohatgi, 1976, pp. 135–136).
•
Exercise 2.119 — P.d.f. of a one-to-one transformation of an absolutely
continuous random vector
Let
• X = (X1 , . . . , Xd ) be an absolutely continuous random vector with joint p.d.f. fX (x);
• Y = (Y1 , . . . , Yd ) = g(X) = AX + b be an invertible affine mapping of IRd into itself,
where A is a nonsingular d × d matrix and b ∈ IRd .
Derive the inverse mapping g −1 and the joint p.d.f. of Y (Karr, 1993, p. 62).
•
Exercise 2.120 — P.d.f. of a one-to-one transformation of an absolutely
continuous random vector
Let
i.i.d.
• X = (X1 , X2 , X3 ) such that Xi ∼ Exponential(1);
X1
X1 +X2
• Y = (Y1 , Y2 , Y3 ) = X1 + X2 + X3 , X1 +X2 +X3 , X1 +X2 .
Derive the joint p.d.f. of Y and conclude that Y1 , Y2 , and Y3 are also independent (Rohatgi,
1976, p. 137).
•
Remark 2.121 — P.d.f. of a one-to-one transformation of an absolutely
continuous random vector (Rohatgi, 1976, p. 136)
In actual applications, we tend to know just k functions, Y1 = g1 (X), . . . , Yk = gk (X).
In this case, we introduce arbitrarily (d − k) (convenient) r.v., Yk+1 = gk+1 (X), . . . , Yd =
gd (X)), such that the conditions of Theorem 2.117 are satisfied.
To find the joint density of the k r.v. we simply integrate the joint p.d.f. fY over all
the (d − k) r.v. that were arbitrarily introduced.
•
We can state a similar result to Theorem 2.117 when g is not a one-to-one
transformation.
98
Theorem 2.122 — P.d.f. of a transformation, with a finite number of inverses,
of an absolutely continuous random vector (Rohatgi, 1976, pp. 136–137)
Assume the conditions of Theorem 2.117 and suppose that:
• for each y ∈ RY ⊂ IRd , the transformation g has a finite number k = k(y) of
inverses;
• RX ⊂ IRd can be partitioned into k disjoint sets, A1 , . . . , Ak , such that the
transformation g from Ai (i = 1, . . . , k) into IRd , say g i , is one-to-one with inverse
transformation g −1
= (g1−1i (y), . . . , gd−1i (y)), i = 1, . . . , k;
i
• the first partial derivatives of g −1
exist, are continuous and that each Jacobian
i
∂g1−1
i (y)
∂y1
Ji (y) =
..
.
∂gd−1
i (y)
∂y1
···
···
···
∂g1−1
i (y)
∂yd
..
.
6= 0,
(2.96)
∂gd−1
i (y)
∂yd
for y = (y1 , . . . , yd ) in the range of the transformation g i .
Then the random vector Y = (Y1 , . . . , Yd ) is absolutely continuous and its joint p.d.f. is
given by
k
h
i
X
fY (y) =
fX g −1
(y)
× |Ji (y)|,
(2.97)
i
i=1
for y = (y1 , . . . , yd ) ∈ RY .
•
Theorem 2.123 — P.d.f. of the sum, difference, product and division of two
absolutely continuous r.v. (Rohatgi, 1976, p. 141)
Let:
• (X, Y ) be an absolutely continuous bidimensional random vector with joint p.d.f.
fX,Y (x, y);
• Z = X + Y , U = X − Y , V = X Y and W = X/Y .
Then
fZ (z) = fX+Y (z)
Z +∞
=
fX,Y (x, z − x) dx
−∞
Z +∞
=
fX,Y (z − y, y) dy
(2.98)
−∞
99
fU (u) = fX−Y (u)
Z +∞
fX,Y (x, x − u) dx
=
−∞
Z +∞
fX,Y (u + y, y) dy
=
(2.99)
−∞
fV (v) = fXY (v)
Z +∞
1
=
fX,Y (x, v/x) ×
dx
|x|
−∞
Z +∞
1
dy
fX,Y (v/y, y) ×
=
|y|
−∞
fW (w) = fX/Y (w)
Z +∞
fX,Y (x, x/w) × |x| dx
=
−∞
Z +∞
fX,Y (wy, y) × |y| dy.
=
(2.100)
(2.101)
−∞
•
Remark 2.124 — P.d.f. of the sum and product of two absolutely continuous
r.v.
It is interesting to note that:
fZ (z) =
=
=
=
=
=
d FZ (z)
dz
d P (Z = X + Y ≤ z)
Z Z dz
d
fX,Y (x, y) dy dx
dz
{(x,y): x+y≤z}
Z +∞ Z z−x
d
fX,Y (x, y) dy dx
dz
−∞
−∞
Z z−x
Z +∞
d
fX,Y (x, y) dy dx
−∞ dz
−∞
Z +∞
fX,Y (x, z − x) dx;
−∞
d FV (v)
dv
d P (V = XY ≤ v)
=
dv
fV (v) =
100
(2.102)
Z Z
d
=
fX,Y (x, y) dy dx
dv
{(x,y): xy≤v}
 R
hR
i
v/x
 +∞ d
f (x, y) dy dx, x > 0
−∞ X,Y
−∞ dv
i
=
R +∞ d hR +∞

f
(x,
y)
dy
dx, x < 0
X,Y
−∞ dv
v/x
Z +∞
1
=
fX,Y (x, v/x) dx.
−∞ |x|
(2.103)
•
Corollary 2.125 — P.d.f. of the sum, difference, product and division of two
independent absolutely continuous r.v. (Rohatgi, 1976, p. 141)
Let:
• X and Y be two independent absolutely continuous r.v. with joint p.d.f.
fX,Y (x, y) = fX (x) × fY (y), ∀x, y;
• Z = X + Y , U = X − Y , V = X Y and W = X/Y .
Then
fZ (z) = fX+Y (z)
Z +∞
fX (x) × fY (z − x) dx
=
−∞
Z +∞
=
fX (z − y) × fY (y) dy
(2.104)
−∞
fU (u) = fX−Y (u)
Z +∞
=
fX (x) × fY (x − u) dx
−∞
Z +∞
=
fX (u + y)fY (y) dy
(2.105)
−∞
fV (v) = fXY (v)
Z +∞
1
=
fX (x) × fY (v/x) ×
dx
|x|
−∞
Z +∞
1
=
fX (v/y) × fY (y) ×
dy
|y|
−∞
fW (w) = fX/Y (w)
Z +∞
=
fX (x) × fY (x/w) × |x| dx
−∞
101
(2.106)
Z
+∞
fX (wy) × fY (y) × |y| dy.
=
(2.107)
−∞
•
Exercise 2.126 — P.d.f. of the sum and difference of two independent
absolutely continuous r.v.
Let X and Y be two r.v. which are independent and uniformly distributed in (0, 1). Derive
the p.d.f. of
(a) X + Y
(b) X − Y
(c) (X + Y, X − Y ) (Rohatgi, 1976, pp. 137–138).
•
Exercise 2.127 — P.d.f. of the mean of two independent absolutely continuous
r.v.
Let X and Y be two independent r.v. with standard normal distribution. Prove that their
∼ Normal(0, 2−1 ).
•
mean X+Y
2
Remark 2.128 — D.f. and p.d.f. of the sum, difference, product and division
of two absolutely continuous r.v.
In several cases it is simpler to obtain the d.f. of those four algebraic functions of X and
Y than to derive the corresponding p.d.f. It suffices to apply Proposition 2.104 and then
differentiate the d.f. to get the p.d.f., as seen in the next exercises.
•
Exercise 2.129 — D.f. and p.d.f. of the difference of two absolutely continuous
r.v.
Choosing adequate underkeel clearance (UKC) is one of the most crucial and most difficult
problems in the navigation of large ships, especially very large crude oil carriers.
Let X be the water depth in a passing shallow waterway, say a harbour or a channel,
and Y be the maximum ship draft. Then the probability of a safe passing a shallow
waterway can be expressed as P (UKC = X − Y > 0).
Assume that X and Y are independent r.v. such that X ∼ Gamma(n, β) and Y ∼
Gamma(m, β), where n, m ∈ IN and m < n. Derive an expression for P (UKC = X − Y >
P
−βx
0) taking into account that FGama(k,β) (x) = ∞
(βx)i /i!, k ∈ IN .
•
i=k e
102
Exercise 2.130 — D.f. and p.d.f. of the sum of two absolutely continuous r.v.
Let X and Y be the durations of two independent system components set in what is called
a stand by connection.11 In this case the system duration is given by X + Y .
Prove that the p.d.f. of X + Y equals
αβ e−βz − e−αz
, z > 0,
fX+Y (z) =
α−β
if X ∼ Exponencial(α) and Y ∼ Exponencial(β), where α, β > 0 and α 6= β.
•
Exercise 2.131 — D.f. of the division of two absolutely continuous r.v.
Let X and Y be the intensity of a transmitted signal and its damping until its reception,
respectively. Moreover, W = X/Y represents the intensity of the received signal.
Assume that the joint p.d.f. of (X, Y ) equals fX,Y (x, y) = λµe−(λx+µy) ×
I(0,+∞)×(0,+∞) (x, y). Prove that the d.f. of W = X/Y is given by:
µ
× I(0,+∞) (w).
(2.108)
FW (w) = 1 −
µ + λw
•
11
At time 0, only the component with duration X is on. The component with duration Y replaces the
other one as soon as it fails.
103
2.5.7
Random variables with prescribed distributions
Motivation 2.132 — Construction of a r.v. with a prescribed distribution (Karr,
1993, p. 63)
Can we construct (or simulate) explicitly individual r.v., random vectors or sequences of
r.v. with prescribed distributions?
•
Proposition 2.133 — Construction of a r.v. with a prescribed d.f. (Karr, 1993,
p. 63)
Let F be a d.f. on IR. Then there is a probability space (Ω, F, P ) and a r.v. X defined
on it such that FX = F .
•
Exercise 2.134 — Construction of a r.v. with a prescribed d.f.
Prove Proposition 2.133 (Karr, 1993, p. 63).
•
The construction of a r.v. with a prescribed d.f. depends on the following definition.
Definition 2.135 — Quantile function (Karr, 1993, p. 63)
The inverse function of F , F −1 , or quantile function associated with F , is defined by
F −1 (p) = inf{x : F (x) ≥ p}, p ∈ (0, 1).
This function is often referred to as the generalized inverse of the d.f.
(2.109)
•
Exercise 2.136 — Quantile functions of an absolutely continuous and a discrete
r.v.
Obtain and draw the graphs of the d.f. and quantile function of:
(a) X ∼ Exponential(λ);
(b) X ∼ Bernoulli(θ).
•
104
Remark 2.137 — Existence of a quantile function (Karr, 1993, p. 63)
Even though F need be neither continuous nor strictly increasing, F −1 always exists.
As the figure of the quantile function (associated with the d.f.) of X ∼ Bernoulli(θ),
F −1 jumps where F is flat, and is flat where F jumps.
Although not necessarily a pointwise inverse of F , F −1 serves that role for many
purposes and has a few interesting properties.
•
Proposition 2.138 — Basic properties of the quantile function (Karr, 1993, p.
63)
Let F −1 be the (generalized) inverse of F or quantile function associated with F . Then
1. For each p and x,
F −1 (p) ≤ x iff p ≤ F (x);
(2.110)
2. F −1 is non decreasing and left-continuous;
3. If F is absolutely continuous, then
F [F −1 (p)] = p, ∀p ∈ (0, 1).
(2.111)
•
Motivation 2.139 — Quantile transformation (Karr, 1993, p. 63)
A r.v. with d.f. F can be constructed by applying F −1 to a r.v. with distribution on (0, 1).
This is usually known as quantile transformation and is a very popular transformation in
random numbers generation/simulation on computer.
•
Proposition 2.140 — Quantile transformation (Karr, 1993, p. 64)
Let F be a d.f. on IR and suppose U ∼ Uniform(0, 1). Then
X = F −1 (U ) has distribution function F.
(2.112)
•
Exercise 2.141 — Quantile transformation
Prove Proposition 2.140 (Karr, 1993, p. 64).
•
Example 2.142 — Quantile transformation
If U ∼ Uniform(0, 1) then both − λ1 ln(1 − U ) and − λ1 ln(U ) have exponential distribution
with parameter λ (λ > 0).
•
105
Remark 2.143 — Quantile transformation (Karr, 1993, p. 64)
R.v. with d.f. F can be simulated by applying F −1 to the (uniformly distributed) values
produced by the random number generator.
Feasibility of this technique depends on either having F −1 available in closed form or
being able to approximate it numerically.
•
Proposition 2.144 — The quantile transformation and the simulation of
discrete and absolutely continuous distributions
To generate (pseudo-)random numbers from a r.v. X with d.f. F , it suffices to:
1. Generate a (pseudo-)random number u from the Uniform(0, 1) distribution.
2. Assign
x = F −1 (u) = inf{m ∈ IR : u ≤ F (m)},
(2.113)
the quantile of order u of X, where F −1 represents the generalized inverse of F . •
For a detailed discussion on (pseudo-)random number generation/generators and their
properties please refer to Gentle (1998, pp. 6–22). For a brief discussion — in Portuguese
— on (pseudo-)random number generation and Monte Carlo simulation method we refer
the reader to Morais (2003, Chapter 2).
Exercise 2.145 — The quantile transformation and the generation of the
Logistic distribution
X is said to have a Logistic(µ, σ) if its p.d.f. is given by
f (x) =
e−
x−µ
σ
, −∞ < x < +∞.
x−µ 2
σ 1 + e− σ
(2.114)
Define the quantile transformation to produce (pseudo-)random numbers with such a
distribution.
•
Exercise 2.146 — The quantile transformation and the simulation of the
Erlang distribution
Describe a method to generate (pseudo-)random numbers from the Erlang(n, λ).12
•
12
Let us remind the reader that the sum of n independent exponential distributions with parameter λ
has an Erlang(n, λ).
106
Exercise 2.147 — The quantile transformation and the generation of the Beta
distribution
Let Y and Z be two independent r.v. with distributions Gamma(α, λ) and Gamma(β, λ),
respectively (α, β, λ > 0).
(a) Prove that X = Y /(Y + Z) ∼ Beta(α, β).
(b) Use this result to describe a random number generation method for the Beta(α, β),
where α, β ∈ IN .
(c) Use any software you are familiar with to generate and plot the histogram of 1000
observations from the Beta(4, 5) distribution.
•
Example 2.148 — The quantile transformation and the generation of the
Bernoulli distribution (Gentle, 1993, p. 47)
To generate (pseudo-)random numbers from the Bernoulli(p) distribution, we should
proceed as follows:
1. Generate a (pseudo-)random number u from the Uniform(0, 1) distribution.
2. Assign
(
x=
0, if u ≤ 1 − p
1, if u > 1 − p
(2.115)
or, equivalently,
(
0, if u ≥ p
x=
1, if u < p.
(2.116)
(What is the advantage of (2.116) over (2.115)?)
•
Exercise 2.149 — The quantile transformation and the simulation of the
Binomial distribution
Describe a method to generate (pseudo-)random numbers from a Binomial(n, p)
distribution.
•
Proposition 2.150 — The converse of the quantile transformation (Karr, 1993,
p. 64)
A converse of the quantile transformation (Propositon 2.140) holds as well, under certain
conditions. In fact, if FX is continuous (not necessarily absolutely continuous) then
FX (X) ∼ Uniform(0, 1).
(2.117)
•
107
Exercise 2.151 — The converse of the quantile transformation
Prove Propositon 2.150 (Karr, 1993, p. 64).
•
Motivation 2.152 — Construction of random vectors with a prescribed
distribution (Karr, 1993, p. 65)
The construction of a random vector with an arbitrary d.f. is more complicated. We shall
address this issue in the next chapter for a special case: when the random vector has
independent components. However, we can state the following result.
•
Proposition 2.153 — Construction of a random vector with a prescribed d.f.
(Karr, 1993, p. 65)
Let F : IRd → [0, 1] be a d−dimensional d.f. Then there is a probability space (Ω, F, P )
•
and a random vector X = (X1 , . . . , Xd ) defined on it such that FX = F .
Motivation 2.154 — Construction of a sequence of r.v. with a prescribed joint
d.f. (Karr, 1993, p. 65)
How to construct a sequence {Xk }k∈IN of r.v. with a prescribed joint d.f. Fn where Fn is
the joint d.f. of X n = (X1 , . . . , Xn ), for each n ∈ IN . The d.f. Fn must satisfy certain
consistency conditions since if such r.v. exists then
Fn (xn ) = P (X1 ≤ x1 , . . . , Xn ≤ xn )
=
lim P (X1 ≤ x1 , . . . , Xn ≤ xn , Xn+1 ≤ x),
x→+∞
(2.118)
•
for all x1 , . . . , xn .
Theorem 2.155 — Kolmogorov existence Theorem (Karr, 1993, p. 65)
Let Fn be a d.f. on IRn , and suppose that
lim Fn+1 (x1 , . . . , xn , x) = Fn (x1 , . . . , xn ),
x→+∞
(2.119)
for each n ∈ IN and x1 , . . . , xn . Then there is a probability space say (Ω, F, P ) and a
sequence of {Xk }k∈IN of r.v. defined on it such that Fn is the d.f. of (X1 , . . . , Xn ), for each
n ∈ IN .
•
Remark 2.156 —
Kolmogorov
existence
Theorem
(http://en.wikipedia.org/wiki/Kolmogorov extension theorem)
Theorem 2.155 guarantees that a suitably “consistent” collection of finitedimensional distributions will define a stochastic process.
This theorem is
credited to soviet mathematician Andrey Nikolaevich Kolmogorov (1903–1987,
http://en.wikipedia.org/wiki/Andrey Kolmogorov).
•
108
References
• Gentle, J.E. (1998). Random Number Generation and Monte Carlo Methods.
Springer-Verlag, New York, Inc. (QA298.GEN.50103)
• Grimmett, G.R. and Stirzaker, D.R. (2001). One Thousand Exercises in Probability.
Oxford University Press.
• Karr, A.F. (1993). Probability. Springer-Verlag.
• Morais, M.C. (2003). Estatı́stica Computacional — Módulo 1: Notas de Apoio
(Caps. 1 e 2), 141 pags.
(http://www.math.ist.utl.pt/∼mjmorais/materialECMCM.html)
• Murteira, B.J.F. (1979). Probabilidades e Estatı́stica (volume I). Editora McGrawHill de Portugal, Lda.
(QA273-280/3.MUR.5922, QA273-280/3.MUR.34472,
QA273-280/3.MUR.34474, QA273-280/3.MUR.34476)
• Resnick, S.I. (1999). A Probability Path. Birkhäuser. (QA273.4-.67.RES.49925)
• Righter, R. (200–). Lectures notes for the course Probability and Risk Analysis
for Engineers. Department of Industrial Engineering and Operations Research,
University of California at Berkeley.
• Rohatgi, V.K. (1976). An Introduction to Probability Theory and Mathematical
Statistics. John Wiley & Sons. (QA273-280/4.ROH.34909)
109
Chapter 3
Independence
Independence is a basic property of events and r.v. in a probability model.
3.1
Fundamentals
Motivation 3.1 — Independence (Resnick, 1999, p. 91; Karr, 1993, p. 71)
The intuitive appeal of independence stems from the easily envisioned property that the
ocurrence of an event has no effect on the probability that an independent event will
occur. Despite the intuitive appeal, it is important to recognize that independence is a
technical concept/definition which must be checked with respect to a specific model.
Independence — or the absence of probabilistic interaction — sets probability apart
as a distinct mathematical theory.
•
A series of definitions of independence of increasingly sophistication will follow.
Definition 3.2 — Independence for two events (Resnick, 1999, p. 91)
Suppose (Ω, F, P ) is a fixed probability space. Events A, B ∈ F are independent if
P (A ∩ B) = P (A) × P (B).
(3.1)
•
Exercise 3.3 — Independence
Let A and B be two independent events. Show that:
(a) Ac and B are independent,
(b) and so are A and B c ,
(c) and Ac and B c .
•
110
Exercise 3.4 — (In)dependence and disjoint events
Let A and B two disjoint events with probabilities P (A), P (B) > 0. Show that these
two events are not independent.
•
Exercise 3.5 — Independence (Exercise 3.2, Karr, 1993, p. 95)
Show that:
(a) an event whose probability is either zero or one is independent of every event;
(b) an event that is independent of itself has probability zero or one.
•
Definition 3.6 — Independence for a finite number of events (Resnick, 1999, p.
91)
The events A1 , . . . , An ∈ F are independent if
!
\
Y
P
Ai =
P (Ai ),
(3.2)
i∈I
i∈I
for all finite I ⊆ {1, . . . , n}.
•
Remark 3.7 — Independence for a finite number of events (Resnick, 1999, p. 92)
P
Note that (3.2) represents nk=2 nk = 2n − n − 1 equations and can be rephrased as
follows:
• the events A1 , . . . , An are independent if
P
n
\
i=1
!
Bi
=
n
Y
P (Bi ),
(3.3)
i=1
where, for each i = 1, . . . , n, Bi equals Ai or Ω.
•
Corollary 3.8 — Independence for a finite number of events (Karr, 1993, p. 81)
Events A1 , . . . , An are independent iff Ac1 , . . . , Acn are independent.
•
Exercise 3.9 — Independence for a finite number of events (Exercise 3.1, Karr,
1993, p. 95)
Let A1 , . . . , An be independent events.
111
S
Q
(a) Prove that P ( ni=1 Ai ) = 1 − ni=1 [1 − P (Ai )].
(b) Consider a parallel system with n components and assume that P (Ai ) is the
reliability of the component i (i = 1, . . . , n). What is the system reliability?
•
Motivation 3.10 — (2nd.) Borel-Cantelli lemma (Karr, 1993, p. 81)
For independent events, Theorem 1.76, the (1st.) Borel-Cantelli lemma, has a converse.
It states that if the events A1 , A2 , . . . are independent and the sum of the probabilities of
the An diverges to infinity, then the probability that infinitely many of them occur is 1. •
Theorem 3.11 — (2nd.) Borel-Cantelli lemma (Karr, 1993, p. 81)
Let A1 , A2 , . . . be independent events. Then
+∞
X
P (An ) = +∞ ⇒ P (lim sup An ) = 1.
(3.4)
n=1
(Moreover, P (lim sup An ) = 1 ⇒
lemma.)
P+∞
n=1
P (An ) = +∞ follows from the 1st. Borel-Cantelli
•
Exercise 3.12 — (2nd.) Borel-Cantelli lemma
Prove Theorem 3.11 (Karr, 1993, p. 82).
•
Definition 3.13 — Independent classes of events (Resnick, 1999, p. 92)
Let Ci ⊆ F, i = 1, . . . , n, be a class of events. Then the classes C1 , . . . , Cn are said to be
independent if for any choice A1 , . . . , An , with Ai ∈ Ci , i = 1, . . . , n, the events A1 , . . . , An
are independent events according to Definition 3.6.
•
Definition 3.14 — Independent σ−algebras (Karr, 1993, p. 94)
Sub σ − algebras G1 , . . . , Gn of σ − algebra F are independent if
!
n
n
\
Y
P
Ai =
P (Ai ),
i=1
(3.5)
i=1
for all Ai ∈ Gi , i = 1, . . . , n.
•
Motivation 3.15 — Independence of σ − algebras (Resnick, 1999, p. 92)
To provide a basic criterion for proving independence of σ −algebras, we need to introduce
the notions of π − system and d − system.
•
112
Definition 3.16 — π−system (Resnick, 1999, p. 32; Karr, 1993, p. 21)
Let P family of subsets of the sample space Ω. P is said to be a π − system if it is closed
under finite intersection: A, B ∈ P ⇒ A ∩ B ∈ P.
•
Remark 3.17 — π−system (Karr, 1993, p. 21)
A σ − algebra is a π − system.
Definition 3.18 — d−system (Karr, 1993, p. 21)
Let D family of subsets of the sample space Ω. D is said to be a d − system
•
1
if it
1. contains the sample space Ω,
2. is closed under proper difference,2
3. and is closed under countable increasing union.3
•
Proposition 3.19 — Relating π− and λ− systems and σ−algebras (Resnick, 1999,
p. 38)
If a class C is both a π − system and λ − system then it is a σ − algebra.
•
Theorem 3.20 — Basic independence criterion (Resnick, 1999, p. 92)
If, for each i = 1, . . . , n, Ci is a non-empty class of events satisfying
1. Ci is a π − system
2. Ci , i = 1, . . . , n are independent
then the σ − algebras generated by these n classes of events, σ(Ci ), . . . , σ(Cn ), are
independent.
•
Exercise 3.21 — Basic independence criterion
Prove the basic independence criterion in Theorem 3.20 (Resnick, 1999, pp. 92–93).
1
Synonyms (Resnick, 1999, p. 36): λ − system, σ − additive, Dynkin class.
If A, B ∈ D and A ⊆ B then B\A ∈ D.
S+∞
3
If A1 ⊆ A2 ⊆ . . . and Ai ∈ D then i=1 Ai ∈ D.
2
113
•
Definition 3.22 — Arbitrary number of independent classes (Resnick, 1999, p.
93; Karr, 1993, p. 94)
Let T be an arbitrary index set. The classes {Ct , t ∈ T } are independent if, for each finite
I such that I ⊂ T , {Ct , t ∈ I} are independent.
An infinite collection of σ − algebras is independent if every finite subcollection is
independent.
•
Corollary 3.23 — Arbitrary number of independent classes (Resnick, 1999, p.
93)
If {Ct , t ∈ T } are non-empty π − systems that are independent then {σ(Ct ), t ∈ T } are
independent.
•
Exercise 3.24 — Arbitrary number of independent classes
Prove Corollary 3.23 by using the basic independence criterion.
114
•
3.2
Independent r.v.
The notion of independence for r.v. can be stated in terms of Borel sets. Moreover, basic
independence criteria can be develloped based solely on intervals such as (−∞, x].
Definition 3.25 — Independence of r.v. (Karr, 1993, p. 71)
R.v. X1 , . . . , Xn are independent if
P ({X1 ∈ B1 , . . . , Xn ∈ Bn }) =
n
Y
P ({Xi ∈ Bi }) ,
(3.6)
i=1
•
for all Borel sets B1 , . . . , Bn .
Independence for r.v. can also be defined in terms of the independence of σ − algebras.
Definition 3.26 — Independence of r.v. (Resnick, 1999, p. 93)
Let T be an arbitrary index set. Then {Xt , t ∈ T } is a family of independent r.v. if
{σ(Xt ), t ∈ T } is a family of independent σ − algebras as stated in Definition 3.22.
•
Remark 3.27 — Independence of r.v. (Resnick, 1999, p. 93)
The r.v. are independent if their induced/generated σ − algebras are independent. The
information provided by any individual r.v. should not affect the probabilistic behaviour
of other r.v. in the family.
Since σ(1A ) = {∅, A, Ac , Ω} we have 1A1 , . . . , 1An independent iff A1 , . . . , An are
independent.
•
Definition 3.28 — Independence of an infinite set of r.v. (Karr, 1993, p. 71)
An infinite set of r.v. is independent if every finite subset of r.v. is independent.
•
Motivation 3.29 — Independence criterion for a finite number of r.v. (Karr,
1993, pp. 71–72)
R.v. are independent iff their joint d.f. is the product of their marginal/individual d.f.
This result affirms the general principle that definitions stated in terms of all Borel
sets need only be checked for intervals (−∞, x].
•
115
Theorem 3.30 — Independence criterion for a finite number of r.v. (Karr, 1993,
p. 72)
R.v. X1 , . . . , Xn are independent iff
n
Y
FX1 ,...,Xn (x1 , . . . , xn ) =
FXi (xi ),
(3.7)
i=1
for all x1 , . . . , xn ∈ IR.
•
Remark 3.31 — Independence criterion for a finite number of r.v. (Resnick,
1999, p. 94)
Theorem 3.30 is usually referred to as factorization criterion.
•
Exercise 3.32 — Independence criterion for a finite number of r.v.
Prove Theorem 3.30 (Karr, 1993, p. 72; Resnick, 1999, p. 94, provides a more
straightforward proof of this result).
•
Theorem 3.33 — Independence criterion for an infinite number of r.v. (Resnick,
1994, p. 94)
Let T be as arbitrary index set. A family of r.v. {Xt , t ∈ T } is independent iff
Y
FXt (xt ),
(3.8)
FI (xt , t ∈ I) =
i∈I
for all finite I ⊂ T and xt ∈ IR.
•
Exercise 3.34 — Independence criterion for an infinite number of r.v.
Prove Theorem 3.33 (Resnick, 1994, p. 94).
•
Specialized criteria for discrete and absolutely continuous r.v. follow from Theorem
3.30.
Theorem 3.35 — Independence criterion for discrete r.v. (Karr, 1993, p. 73;
Resnick, 1999, p. 94)
The discrete r.v. X1 , . . . , Xn , with countable ranges R1 , . . . , Rn , are independent iff
n
Y
P ({X1 = x1 , . . . , Xn = xn }) =
P ({Xi = xi }),
(3.9)
i=1
for all xi ∈ Ri , i = 1, . . . , n.
•
Exercise 3.36 — Independence criterion for discrete r.v.
Prove Theorem 3.35 (Karr, 1993, p. 73; Resnick, 1999, pp. 94–95).
•
116
Exercise 3.37 — Independence criterion for discrete r.v.
The number of laptops (X) and PCs (Y ) sold daily in a store have a joint p.f. partially
described in the following table:
X
0
1
2
Y
0
1
0.1 0.1
0.2 0.1
0 0.1
2
0.3
0.1
a
Complete the table and prove that X and Y are not independent r.v.
•
Theorem 3.38 — Independence criterion for absolutely continuous r.v. (Karr,
1993, p. 74)
Let X = (X1 , . . . , Xn ) be an absolutely continuous random vector. Then X1 , . . . , Xn are
independent iff
fX1 ,...,Xn (x1 , . . . , xn ) =
n
Y
fXi (xi ),
(3.10)
i=1
for all x1 , . . . , xn ∈ IR.
•
Exercise 3.39 — Independence criterion for absolutely continuous r.v.
Prove Theorem 3.38 (Karr, 1993, p. 74).
•
Exercise 3.40 — Independence criterion for absolutely continuous r.v.
The r.v. X and Y represent the lifetimes (in 103 hours) of two components of a control
system and have joint p.d.f. given by
(
1, 0 < x < 1, 0 < y < 1
fX,Y (x, y) =
(3.11)
0, otherwise.
•
Prove that X and Y are independent r.v.
Exercise 3.41 — Independence criterion for absolutely continuous r.v.
Let X and Y be two r.v. that represent, respectively, the width (in dm) and the length
(in dm) of a rectangular piece. Admit the joint p.d.f. of (X, Y ) is given by
(
2, 0 < x < y < 1
fX,Y (x, y) =
(3.12)
0, otherwise.
Prove that X and Y are not independent r.v.
117
•
Example 3.42 — Independent r.v. (Karr, 1993, pp. 75–76)
Independent r.v. are inherent to certain probability structures.
• Binary expansions4
Let P be the uniform distribution on Ω = [0, 1]. Each point ω ∈ Ω has a binary
expansion
ω → 0. X1 (ω) X2 (ω) . . . ,
(3.13)
where the Xi are functions from Ω to {0, 1}.
This expansion is “unique” and it can be shown that X1 , X2 , . . . are independent
P
−n
Xn ∼ Uniform(0, 1).
and with a Bernoulli(p = 21 ) distribution.5 Moreover, +∞
n=1 2
According to Resnick (1999, pp. 98-99), the binary expansion of 1 is 0.111. . . since
+∞
X
2−n × 1 = 1.
(3.14)
n=1
In addition, if a number such a 12 has two possible binary expansions, we agree to
use the non terminating one. Thus, even though 12 has two expansions 0.0111. . .
and 0.1000. . . because
−1
2
×0+
+∞
X
2−n × 1 =
1
2
(3.15)
2−n × 0 =
1
,
2
(3.16)
n=2
2−1 × 1 +
+∞
X
n=2
by convention, we use the first binary expansion.
• Multidimensional uniform distribution
Suppose that P is the uniform distribution on [0, 1]n . Then the coordinate r.v.
Ui ((ω1 , . . . , ωn )) = ωi , i = 1, . . . , n, are independent, and each of them is uniformly
distributed on [0, 1]. In fact, for intervals I1 , . . . , In ,
P ({U1 ∈ I1 , . . . , Un ∈ In }) =
n
Y
P ({Ui ∈ Ii }) .
i=1
4
5
Or dyadic expansions of uniform random numbers (Resnick, 1999, pp. 98-99).
The proof of this result can also be found in Resnick (1999, pp. 99-100).
118
(3.17)
In other cases, whether r.v. are independent depends on the value of a parameter.
• Bivariate normal distribution
Let (X, Y ) be a random vector with a bivariate normal distribution with p.d.f.
x2 − 2ρxy + y 2
, (x, y) ∈ IR2
fX,Y (x, y) = p
exp −
2)
2
2(1
−
ρ
2π 1 − ρ
1
(3.18)
X and Y have both marginal standard normal distributions then, by the
factorization criterion, X and Y are independent iff ρ = 0.
•
Exercise 3.43 — Bivariate normal distributed r.v. (Karr, 1993, p. 96, Exercise
3.8)
Let (X, Y ) have the bivariate normal p.d.f.
2
1
x − 2ρxy + y 2
, (x, y) ∈ IR2 .
(3.19)
exp −
fX,Y (x, y) = p
2)
2
2(1
−
ρ
2π 1 − ρ
(a) Prove that X ∼ Y ∼ normal(0, 1).
(b) Prove that X and Y are independent iff ρ = 0.
(c) Calculate P ({X ≥ 0, Y ≥ 0}) as a function of ρ.
•
Exercise 3.44 — I.i.d. r.v. with absolutely continuous distributions (Karr, 1993,
p. 96, Exercise 3.9)
Let (X, Y ) be an absolutely continuous random vector where X and Y are i.i.d. r.v. with
absolutely continuous d.f. F . Prove that:
a P ({X = Y }) = 0;
b P ({X < Y }) = 12 .
•
119
3.3
Functions of independent r.v.
Motivation 3.45 — Disjoint blocks theorem (Karr, 1993, p. 76)
R.v. that are functions of disjoint subsets of a family of independent r.v. are also
independent.
•
Theorem 3.46 — Disjoint blocks theorem (Karr, 1993, p. 76)
Let:
• X1 , . . . , Xn be independent r.v.;
• J1 , . . . , Jk be disjoint subsets of {1, . . . , n};
• Yl = gl X (l) , where gl is a Borel measurable function and X (l) = {Xi , i ∈ Jl } is a
subset of the family of the independent r.v., for each l = 1, . . . , k.
Then
Y1 = g1 X (1) , . . . , Yk = gk X (k)
(3.20)
•
are independent r.v.
Remark 3.47 — Disjoint blocks theorem (Karr, 1993, p. 77)
According to Definitions 3.25 and 3.28, the disjoint blocks theorem can be extended to
(countably) infinite families and blocks.
•
Exercise 3.48 — Disjoint blocks theorem
Prove Theorem 3.46 (Karr, 1993, pp. 76–77).
•
Example 3.49 — Disjoint blocks theorem
Let X1 , . . . , X5 be five independent r.v., and J1 = {1, 2} and J2 = {3, 4} two disjoint
subsets of {1, . . . , 5}. Then
• Y1 = X1 + X2 = g1 (Xi , i ∈ J1 = {1, 2}) and
• Y2 = X3 − X4 = g2 (Xi , i ∈ J2 = {3, 4})
•
are independent r.v.
120
Corollary 3.50 — Disjoint blocks theorem (Karr, 1993, p. 77)
Let:
• X1 , . . . , Xn be independent r.v.;
• Yi = gi (Xi ), i = 1, . . . , n, where g1 , . . . , gn are (Borel measurable) functions from IR
to IR.
•
Then Y1 , . . . , Yn are independent r.v.
We have already addressed the p.d.f. (or p.f.) of a sum, difference, product or
division of two independent absolutely continuous (or discrete) r.v. However, the sum
of independent absolutely continuous r.v. merit special consideration — its p.d.f. has a
specific designation: convolution of p.d.f..
Definition 3.51 — Convolution of p.d.f. (Karr, 1993, p. 77)
Let:
• X and Y be two independent absolutely continuous r.v.;
• f and g be the p.d.f. of X and Y , respectively.
Then the p.d.f. of X + Y is termed the convolution of the p.d.f. f and g, represented by
f ? g and given by
Z +∞
f (t − s) × g(s) ds.
(3.21)
(f ? g)(t) =
−∞
•
Proposition 3.52 — Properties of the convolution of p.d.f. (Karr, 1993, p. 78)
The convolution of d.f. is:
• commutative — f ? g = g ? f , for all p.d.f. f and g;
• associative — (f ? g) ? h = f ? (g ? h), for all p.d.f. f , g and h.
Exercise 3.53 — Convolution of p.f.
How could we define the convolution of the p.f. of two independent discrete r.v.?
121
•
•
Exercise 3.54 — Sum of independent binomial distributions
Let X ∼ Binomial(nX , p) and Y ∼ Binomial(nY , p) be independent.
Prove that X + Y ∼ Binomial(nX + nY , p) by using the Vandermonde’s identity
•
(http://en.wikipedia.org/wiki/Vandermonde’s identity).6
Exercise 3.54 gives an example of a distribution family which is closed under
convolution. There are several other families with the same property, as illustrated by
the next proposition.
Proposition 3.55 — A few distribution families closed under convolution
R.v.
Xi ∼indep NegativeBinomial(ri , p), i = 1, . . . , n
Convolution
P
Pk
k
i=1 Xi ∼ Binomial
i=1 ni , p
P
Pn
k
i=1 Xi ∼ NegativeBinomial
i=1 ri , p
Xi ∼indep Poisson(λi ), i = 1, . . . , n
Pn
Xi ∼indep Binomial(ni , p), i = 1, . . . , k
Xi ∼indep. Normal(µi , σi2 ), i = 1, . . . , n
Xi ∼indep. Gamma(αi , λ), i = 1, . . . , n
Pn
Xi ∼ Poisson ( i=1 λi )
Pn
Pn
Pn
2
i=1 Xi ∼ Normal
i=1 µi ,
i=1 σi
Pn
Pn
Pn 2 2 i=1 ci Xi ∼ Normal
i=1 ci µi ,
i=1 ci σi
Pn
Pn
i=1 Xi ∼ Gamma (
i=1 αi , λ)
i=1
•
Exercise 3.56 — Sum of (in)dependent normal distributions
Let (X, Y ) have a (non-singular) bivariate normal distribution with mean vector and
covariance matrix
#
"
#
"
2
ρσX σY
µX
σX
µ=
and Σ =
,
(3.22)
µY
ρσX σY
σY2
respectively, that is, the joint p.d.f. is given by
(
"
2
1
1
x − µX
p
fX,Y (x, y) =
exp −
2(1 − ρ2 )
σX
2πσX σY 1 − ρ2
2 #)
x − µX
y − µY
y − µY
−2ρ
+
, (x, y) ∈ IR2 ,
σX
σY
σY
6
(3.23)
(3.24)
In combinatorial mathematics, Vandermonde’s identity, named after Alexandre-Théophile
n Pr
m
Vandermonde (1772), states that the equality m+n
=
k=0 k
r
r−k , m, n, r ∈ IN0 , for binomial
coefficients holds. This identity was given already in 1303 by the Chinese mathematician Zhu Shijie (Chu
Shi-Chieh).
122
for |ρ| = |corr(X, Y )| < 1.7
Prove that X + Y is normally distributed with parameters E(X + Y ) = µX + µY and
2
V (X + Y ) = V (X) + 2cov(X, Y ) + V (Y ) = σX
+ 2ρσX σY + σY2 .
•
Exercise 3.57 — Distribution of the minimum of two exponentially distributed
r.v. (Karr, 1993, p. 96, Exercise 3.7)
Let X ∼ Exponential(λ) and Y ∼ Exponential(µ) be two independent r.v.
Calculate the distribution of Z = min{X, Y }.
•
Exercise 3.58 — Distribution of the minimum of exponentially distributed r.v.
i.i.d.
Let X ∼ Exponential(λi ) and ai > 0, for i = 1,. . . , n. Pn λi
•
Prove that mini=1,...,n {ai Xi } ∼ Exponential
i=1 ai .
Exercise 3.59 — Distribution of the minimum of Pareto distributed r.v.
The Pareto distribution, named after the Italian economist Vilfredo Pareto, was originally
used to model the wealth of individuals, X.8
We say that X ∼ Pareto(b, α) if
α bα
(3.25)
fX (x) = α+1 , x ≥ b,
x
where b > 0 is the minimum possible value of X (it also represents the scale parameter)
and α > 0 is called the Pareto index (or the shape parameter)
i.i.d.
Consider n individuals with wealths Xi ∼ X, i = 1, . . . , n. Identify the survival
function of the minimal wealth of these n individuals and comment on the result.
•
Proposition 3.60 — A few distribution families closed under the minimum
operation
R.v.
Minimum
Xi ∼indep Geometric(pi ), i = 1, . . . , n
Xi ∼indep Exponential(λi ), ai > 0, i = 1, . . . , n
Qn
mini=1,...,n Xi ∼ Geometric (1 − i=1 (1 − pi ))
P
n
λi
mini=1,...,n ai Xi ∼ Exponential
i=1 ai
Xi ∼indep Pareto(b, αi ), i = 1, . . . , n, a > 0
mini=1,...,n aXi ∼ Pareto (ab,
Pn
i=1
αi )
•
7
The fact that two random variables X and Y both have a normal distribution does not imply that
the pair (X, Y ) has a joint normal distribution. A simple example is one in which Y = X if |X| > 1
and Y = −X if |X| < 1. This is also true for more than two random variables. (For more details see
http://en.wikipedia.org/wiki/Multivariate normal distribution).
8
The Pareto distribution seemed to show rather well the way that a larger portion of
the wealth of any society is owned by a smaller percentage of the people in that society
(http://en.wikipedia.org/wiki/Pareto distribution).
123
Exercise 3.61 — A few distribution families closed under the minimum
operation
Prove the first and the third results of Proposition 3.60
Can we generalize the closure property of the family of Pareto distributions? (Hint: b
is a scale parameter.)
•
124
3.4
Order statistics
Algebraic operations on independent r.v., such as the minimum, the maximum and order
statistics, are now further discussed because they play a major role in applied areas such
as reliability.
Definition 3.62 — System reliability function (Barlow and Proschan, 1975, p. 82)
The system reliability function for the interval [0, t] is the probability that the system
functions successfully throughout the interval [0, t].
If T represents the system lifetime then the system reliability function is the survival
function of T ,
ST (t) = P ({T > t}) = 1 − FT (t).
(3.26)
If the system has n components with independent lifetimes X1 , . . . , Xn , with survival
functions SX1 (t), . . . , SXn (t), then system reliability function is a function of those n
reliability functions, i.e,
ST (t) = h [SX1 (t), . . . , SXn (t)] .
(3.27)
If they are not independent then ST (t) depends on more than the component marginal
distributions at time t.
•
Definition 3.63 — Order statistics
Given any r.v., X1 , X2 , . . . , Xn ,
• the 1st. order statistic is the minimum of X1 , . . . , Xn , X(1) = mini=1,...,n Xi ,
• nth. order statistic is the maximum of X1 , . . . , Xn , X(n) = maxi=1,...,n Xi , and
• the ith. order statistic corresponds to the ith.-smallest r.v. of X1 , . . . , Xn , X(i) ,
i = 1, . . . , n.
Needless to say that the order statistics X(1) , X(2) , . . . , X(n) are also r.v., defined by sorting
X1 , X2 , . . . , Xn in increasing order. Thus, X(1) ≤ X(2) ≤ . . . ≤ X(n) .
•
Motivation 3.64 — Importance of order statistics in reliabilty
A system lifetime T can be expressed as a function of order statistics of the components
lifetimes, X1 , . . . , Xn .
i.i.d.
If we assume that Xi ∼ X, i = 1, . . . , n, then the system reliability function ST (t) =
P ({T > t}) can be easily written in terms of the survival function (or reliability function)
of X, SX (t) = P ({X > t}), for some of the most usual reliability structures.
•
125
Example 3.65 — Reliability function of a series system
A series system functions if all its components function. Therefore the system lifetime is
given by
T = min{X1 , . . . , Xn } = X(1) .
(3.28)
i.i.d.
If Xi ∼ X, i = 1, . . . , n, then the system reliability function is defined as
!
n
\
ST (t) = P
{Xi > t}
i=1
= [SX (t)]n ,
(3.29)
•
where SX (t) = P ({X > t}).
Exercise 3.66 — Reliability function of a series system
A series system has two components with i.i.d. lifetimes with hcommon failurei rate function
Rt
= 0.5t−0.5 , t ≥ 0, i.e., SX (t) = exp − 0 λX (s) ds . (Prove this
given by λX (t) = SfXX (t)
(t)
result!).
Derive the system reliability function.
•
Example 3.67 — Reliability function of a parallel system
A parallel system functions if at least one of its components functions. Therefore the
system lifetime is given by
T = max{X1 , . . . , Xn } = X(n) .
(3.30)
i.i.d.
If Xi ∼ X, i = 1, . . . , n, then the system reliability function equals
ST (t) = 1 − FT (t)
!
n
\
{Xi ≤ t}
= 1−P
i=1
= 1 − [1 − SX (t)]n ,
(3.31)
•
where SX (t) = P ({X > t}).
Exercise 3.68 — Reliability function of a parallel system
An obsolete electronic system has 6 valves set in parallel. Assume that the components
2
lifetime (in years) are i.i.d. r.v. with common p.d.f. fX (t) = 50 t e−25t , t > 0.
Obtain the system reliability for 2 months.
•
126
Proposition 3.69 — Joint p.d.f. of the order statistics and more (Murteira, 1980,
pp. 57, 55, 54)
i.i.d.
Let X1 , . . . , Xn be absolutely continuous r.v. such that Xi ∼ X, i = 1, . . . , n. Then:
fX(1) ,...,X(n) (x1 , . . . , xn ) = n! ×
n
Y
fX (xi ),
(3.32)
i=1
for x1 ≤ . . . ≤ xn ;
FX(i) (x) =
n X
n
j=i
j
× [FX (x)]j × [1 − FX (x)]n−j
= 1 − FBinomial(n,FX (x)) (i − 1),
(3.33)
for i = 1, . . . , n;
fX(i) (x) =
n!
× [FX (x)]i−1 × [1 − FX (x)]n−i × fX (x),
(i − 1)! (n − i)!
(3.34)
for i = 1, . . . , n;
fX(i) ,X(j) (xi , xj ) =
n!
(i − 1)! (j − i − 1)! (n − j)!
× [FX (xi )]i−1 × [FX (xj ) − FX (xi )]j−i−1 × [1 − FX (xj )]n−j
×fX (xi ) × fX (xj ),
(3.35)
for xi < xj , and 1 ≤ i < j ≤ n.
•
Exercise 3.70 — Joint p.d.f. of the order statistics and more
Prove Proposition 3.69 (http://en.wikipedia.org/wiki/Order statistic).
•
Example 3.71 — Reliability function of a k-out-of-n system
A k-out-of-n system functions if at least k out of its n components function. A series
system corresponds to a n-out-of-n system, whereas a parallel system corresponds to a
1-out-of-n system. The lifetime of a k-out-of-n system is also associated to an order
statistic:
T = X(n−k+1) .
(3.36)
i.i.d.
If Xi ∼ X, i = 1, . . . , n, then the system reliability function can also be derived by using
the auxiliary r.v.
Zt = number of Xi0 s > t ∼ Binomial(n, SX (t)).
127
(3.37)
In fact,
ST (t) = P (Zt ≥ k)
= 1 − P (Zt ≤ k − 1)
= 1 − FBinomial(n,SX (t)) (k − 1)
= P (n − Zt ≤ n − k)
= FBinomial(n,FX (t)) (n − k).
(3.38)
•
Exercise 3.72 — Reliability function of a k-out-of-n system
Admit a machine has 4 engines and it only functions if at least 3 of those engines are
working. Moreover, suppose the lifetimes of the engines (in thousand hours) are i.i.d. r.v.
with Exponential distribution with scale parameter λ−1 = 2.
Obtain the machine reliability for a period of 1000 h.
•
128
3.5
Constructing independent r.v.
The following theorem is similar to Proposition 2.133 and guarantees that we can also
construct independent r.v. with prescribed d.f.
Theorem 3.73 — Construction of a finite collection of independent r.v. with
prescribed d.f. (Karr, 1993, p. 79)
Let F1 , . . . , Fn be d.f. on IR. Then there is a probability space (Ω, F, P ) and r.v.
X1 , . . . , Xn defined on it such that X1 , . . . , Xn are independent r.v. and FXi = Fi for
each i.
•
Exercise 3.74 — Construction of a finite collection of independent r.v. with
prescribed d.f.
Prove Proposition 3.73 (Karr, 1993, p. 79).
•
Theorem 3.75 — Construction of a sequence collection of independent r.v.
with prescribed d.f. (Karr, 1993, p. 79)
Let F1 , F2 , . . . be d.f. on IR. Then there is a probability space (Ω, F, P ) and a r.v.
X1 , X2 , . . . defined on it such that X1 , X2 , . . . are independent r.v. and FXi = Fi for
each i.
•
Exercise 3.76 — Construction of a sequence collection of independent r.v. with
prescribed d.f.
Prove Proposition 3.75 (Karr, 1993, pp. 79-80).
•
129
3.6
Bernoulli process
Motivation 3.77 — Bernoulli (counting) process (Karr, 1993, p. 88)
Counting sucesses in repeated, independent trials, each of which has one of two possible
outcomes (success 9 and failure).
•
Definition 3.78 — Bernoulli process (Karr, 1993, p. 88)
A Bernoulli process with parameter p is a sequence {Xi , i ∈ IN } of i.i.d. r.v. with Bernoulli
distribution with parameter p = P (success).
•
Definition 3.79 — Important r.v. in a Bernoulli process (Karr, 1993, pp. 88–89)
In isolation a Bernoulli process is neither deep or interesting. However, we can identify
three associated and very important r.v.:
• Sn =
Pn
i=1
Xi , the number of successes in the first n trials (n ∈ IN );
• Tk = min{n : Sn = k}, the time (trial number) at which the kth. success occurs
(k ∈ IN ), that is, the number of trials needed to get k successes;
• Uk = Tk −Tk−1 , the time (number of trials) between the kth. and (k −1)th. successes
(k ∈ IN, T0 = 0, U1 = T1 ).
•
Definition 3.80 — Bernoulli counting process (Karr, 1993, p. 88)
The sequence {Sn , n ∈ IN } is usually termed as Bernoulli counting process (or success
counting process).
•
Exercise 3.81 — Bernoulli counting process
Simulate a Bernoulli process with parameter p = 21 and consider n = 100 trials. Plot the
realizations of both the Bernoulli process and the Bernoulli counting process.
•
Definition 3.82 — Bernoulli success time process (Karr, 1993, p. 88)
The sequence {Tk , k ∈ IN } is usually called the Bernoulli success time process.
9
Or arrival.
130
•
Proposition 3.83 — Important distributions in a Bernoulli process (Karr, 1993,
pp. 89–90)
In a Bernoulli process with parameter p (p ∈ [0, 1]) we have:
• Sn ∼ Binomial(n, p), n ∈ IN ;
• Tk ∼ NegativeBinomial(k, p), k ∈ IN ;
i.i.d.
d
• Uk ∼ Geometric(p) = NegativeBinomial(1, p), k ∈ IN .
•
Exercise 3.84 — Bernoulli counting process
i.i.d.
(a) Prove that Tk ∼ NegativeBinomial(k, p) and Uk ∼ Geometric(p), for k ∈ IN .
(b) Consider a Bernoulli process with parameter p = 1/2 and obtain the probability of
having 57 successes between time 10 and 100.
•
Exercise 3.85 — Relating the Bernoulli counting process and random walk
(Karr, 1993, p. 97, Exercise 3.21)
Let Sn be a Bernoulli (counting) process with p = 21 .
Prove that the process Zn = 2Sn − n is a symmetric random walk.
•
Proposition 3.86 — Properties of the Bernoulli counting process (Karr, 1993,
p. 90)
The Bernoulli counting process {Sn , n ∈ IN } has:
• independent increments — i.e., for 0 < n1 < . . . < nk , the r.v. Sn1 , Sn2 − Sn1 ,
Sn3 − Sn2 , . . ., Snk − Snk−1 are independent;
• stationary increments — that is, for fixed j, the distribution of Sk+j − Sk is the
same for all k ∈ IN .
•
Exercise 3.87 — Properties of the Bernoulli counting process
Prove Proposition 3.86 (Karr, 1993, p. 90).
131
•
Remark 3.88 — Bernoulli counting process (web.mit.edu/6.262/www/lectures/
6.262.Lec1.pdf)
Some application areas for discrete stochastic processes such as the Bernoulli counting
process (and the Poisson process, studied in the next section) are:
• Operations Research
Queueing in any area, failures in manufacturing systems, finance, risk modelling,
network models
• Biology and Medicine
Epidemiology, genetics and DNA studies, cell modelling, bioinformatics, medical
screening, neurophysiology
• Computer Systems
Communication networks, intelligent control systems, data compression, detection
of signals, job flow in computer systems, physics – statistical mechanics.
•
Exercise 3.89 — Bernoulli process modelling of sexual HIV transmission
(Pinkerton and Holtgrave (1998, pp. 13–14))
In the Bernoulli-process model of sexual HIV transmission, each act of sexual intercourse
is treated as an independent stochastic trial that is associated to a probability α of HIV
transmission. α is also known as the infectivity of HIV and a number of factors are
believed to influence α.10
(a) Prove that the expression of the probability of HIV transmission in n multiple
contacts with the same infected partner is 1 − (1 − α)n .
(b) Assume now that the consistent use of condoms reduce the infectivity from α to
α0 = (1 − 0.9) × α.11 Derive the relative change reduction in the probability defined
in (a) due to the consistent use of condoms. Evaluate this reduction when α = 0.01
and n = 10.
•
Definition 3.90 — Independent Bernoulli processes
(1)
(2)
Two Bernoulli counting processes {Sn , n ∈ IN } and {Sn , n ∈ IN } are independent
if for every positive integer k and all times n1 , . . . , nk , we have that
the random
(1)
(1)
(2)
(2)
vector Sn1 , . . . , Snk associated with the first process is independent of Sn1 , . . . , Snk
associated with the second process.
•
10
Such as the type of sex act engaged, sex role, etc.
α is termed reduced infectivity; 0.9 represents a conservative estimate of condom effectiveness.
11 0
132
Proposition 3.91 — Merging independent Bernoulli processes
(1)
(2)
Let {Sn , n ∈ IN } and {Sn , n ∈ IN } be two independent Bernoulli counting processes
(1)
(2)
with parameters α and β, respectively. Then the merged process {Sn ⊕ Sn , n ∈ IN } is
a Bernoulli counting process with parameter α + β − αβ.
•
Exercise 3.92 — Merging independent Bernoulli processes
Prove Proposition 3.91.
•
Proposition 3.93 — Splitting a Bernoulli process (or sampling a Bernoulli
process)
Let {Sn , n ∈ IN } be a Bernoulli counting process with parameter α. Splitting the original
Bernoulli counting process based on a selection probability p yields two Bernoulli counting
processes with parameters αp and α(1 − p).
•
Exercise 3.94 — Splitting a Bernoulli process
Prove Proposition 3.91.
Are the two resulting processes independent?12
12
•
NO! If we try to merge the two splitting processes and assume they are independent we get a
parameter αp + α(1 − p) − αp × α(1 − p) which is different form α.
133
3.7
Poisson process
In what follows we use the notation of Ross (1989, Chapter 5) which is slightly different
from the one of Karr (1993, Chapter 3).
Motivation 3.95 — Poisson process (Karr, 1993, p. 91)
Is there a continuous analogue of the Bernoulli process? yes!
The Poisson process, named after the French mathematician Siméon-Denis Poisson
(1781–1840), is the stochastic process in which events occur continuously and
independently of one another. Examples that are well-modeled as Poisson processes
include the radioactive decay of atoms, telephone calls arriving at a switchboard, and
page view requests to a website.13
•
Definition 3.96 — Counting process (in continuous time) (Ross, 1989, p. 209)
A stochastic process {N (t), t ≥ 0} is said to a counting process if N (t) represents the
total number of events (e.g. arrivals) that have occurred up to time t. From this definition
we can conclude that a counting process {N (t), t ≥ 0} must satisfy:
• N (t) ∈ IN0 , ∀ t ≥ 0;
• N (s) ≤ N (t), ∀ 0 ≤ s < t;
• N (t) − N (s) corresponds to the number of events that have occurred in the interval
(s, t], ∀ 0 ≤ s < t.
•
Definition 3.97 — Counting process (in continuous time) with independent
increments (Ross, 1989, p. 209)
The counting process {N (t), t ≥ 0} is said to have independent increments if the number
of events that occur in disjoint intervals are independent r.v., i.e.,
• for 0 < t1 < . . . < tn , N (t1 ), N (t2 ) − N (t1 ), N (t3 ) − N (t2 ), . . ., N (tn ) − N (tn−1 ) are
independent r.v.
•
13
For more examples check http://en.wikipedia.org/wiki/Poisson process.
134
Definition 3.98 — Counting process (in continuous time) with stationary
increments (Ross, 1989, p. 210)
The counting process {N (t), t ≥ 0} is said to have stationary increments if distribution
of the number of events that occur in any interval of time depends only on the length of
the interval,14 that is,
d
• N (t2 + s) − N (t1 + s) = N (t2 ) − N (t1 ), ∀ s > 0, 0 ≤ t1 < t2 .
•
Definition 3.99 — Poisson process (Karr, 1993, p. 91)
A counting process {N (t), t ≥ 0} is said to be a Poisson process with rate λ (λ > 0) if:
• {N (t), t ≥ 0} has independent and stationary increments;
• N (t) ∼ Poisson(λt).
•
Remark 3.100 — Poisson process (Karr, 1993, p. 91)
Actually, N (t) ∼ Poisson(λt) follows from the fact that {N (t), t ≥ 0} has independent
and stationary increments, thus, redundant in Definition 3.99.
•
Definition 3.101 — The definition of a Poisson process revisited (Ross, 1989, p.
212)
The counting process {N (t), t ≥ 0} is said to be a Poisson process with rate λ, if
• N (0) = 0;
• {N (t), t ≥ 0} has independent and stationary increments;
• P ({N (h) = 1}) = λh + o(h);15
• P ({N (h) ≥ 2}) = o(h).
•
Exercise 3.102 — The definition of a Poisson process revisited
Prove that Definitions 3.99 and 3.101 are equivalent (Ross, 1989, pp. 212–214).
14
The distributions do not depend on the origin of the time interval; they only depend on the length
of the interval.
15
The function f is said to be o(h) if limh→0 f (h)
h = 0 (Ross, 1989, p. 211).
f (h)
2
The function f (x) = x is o(h) since limh→0 h = limh→0 h = 0.
The function f (x) = x is not o(h) since limh→0 f (h)
h = limh→0 1 = 1 6= 0.
135
Proposition 3.103 — Joint p.f. of N (t1 ), . . . , N (tn ) in a Poisson process (Karr,
1993, p. 91)
For 0 < t1 < . . . < tn and 0 ≤ k1 ≤ . . . ≤ kn ,
P ({N (t1 ) = k1 , . . . , N (tn ) = kn }) =
n
Y
e−λ(tj −tj−1 ) [λ(tj − tj−1 )]kj −kj−1
.
(k
−
k
)!
j
j−1
j=1
(3.39)
•
Exercise 3.104 — Joint p.f. of N (t1 ), . . . , N (tn ) in a Poisson process
Prove Proposition 3.103 (Karr, 1993, p. 92) by taking advantage, namely, of the fact that
a Poisson process has independent increments.
•
Exercise 3.105 — Joint p.f. of N (t1 ), . . . , N (tn ) in a Poisson process (“Stochastic
Processes” — Test of 2002-11-09)
A machine produces electronic components according to a Poisson process with rate equal
to 10 components per hour. Let N (t) be the number of produced components up to time
t.
Evaluate the probability of producing at least 8 components in the first hour given
that exactly 20 components have been produced in the first two hours.
•
Definition 3.106 — Important r.v. in a Poisson process (Karr, 1993, pp. 88–89)
Let {N (t), t ≥ 0} be a Poisson process with rate λ. Then:
• Sn = inf{t : N (t) = n} represents the time of the occurrence of the nth. event (e.g.
arrival), n ∈ IN ;
• Xn = Sn − Sn−1 corresponds to the time between the nth. and (n − 1)th. events
(e.g. interarrival time), n ∈ IN .
•
136
Proposition 3.107 — Important distributions in a Poisson process (Karr, 1993,
pp. 92–93)
So far we know that N (t) ∼ Poisson(λt), t > 0. We can also add that:
• Sn ∼ Erlang(n, λ), n ∈ IN ;
i.i.d.
• Xn ∼ Exponential(λ), n ∈ IN .
•
Remark 3.108 — Relating N (t) and Sn in a Poisson process
We ought to note that:
N (t) ≥ n ⇔ Sn ≤ t
(3.40)
FSn (t) = FErlang(n,λ) (t)
= P ({N (t) ≥ n})
+∞ −λt
X
e (λt)j
=
j!
j=n
= 1 − FP oisson(λt) (n − 1), n ∈ IN.
(3.41)
•
Exercise 3.109 — Important distributions in a Poisson process
Prove Proposition 3.107 (Karr, 1993, pp. 92–93).
•
Exercise 3.110 — Time between events in a Poisson process
Suppose that people immigrate into a territory at a Poisson rate λ = 1 per day.
What is the probability that the elapsed time between the tenth and the eleventh
arrival exceeds two days? (Ross, 1989, pp. 216–217).
•
Exercise 3.111 — Poisson process
Simulate a Poisson process with rate λ = 1 considering the interval [0, 100]. Plot the
realizations of the Poisson process.
The sample path of a Poisson process should look like this:
•
137
Motivation 3.112 — Conditional distribution of the first arrival time (Ross,
1989, p. 222)
Suppose we are told that exactly one event of a Poisson process has taken place by time
t (i.e. N (t) = 1), and we are asked to determine the distribution of the time at which the
event occurred (S1 ).
•
Proposition 3.113 — Conditional distribution of the first arrival time (Ross,
1989, p. 223)
Let {N (t), t ≥ 0} be a Poisson process with rate λ > 0. Then
S1 |{N (t) = 1} ∼ Uniform(0, t).
(3.42)
•
Exercise 3.114 — Conditional distribution of the first arrival time
Prove Proposition 3.113 (Ross, 1989, p. 223).
•
Proposition 3.115 — Conditional distribution of the arrival times (Ross, 1989,
p. 224)
Let {N (t), t ≥ 0} be a Poisson process with rate λ > 0. Then
fS1 ,...,Sn |{N (t)=n} (s1 , . . . , sn ) =
n!
,
tn
(3.43)
for 0 < s1 < . . . < sn < t and n ∈ IN .
•
Remark 3.116 — Conditional distribution of the arrival times (Ross, 1989, p.
224)
Proposition 3.115 is usually paraphrased as stating that, under the condition that n events
have occurred in (0, t), the times S1 , . . . , Sn at which events occur, considered as unordered
r.v., are i.i.d. and Uniform(0, t).16
•
Exercise 3.117 — Conditional distribution of the arrival times
Prove Proposition 3.115 (Ross, 1989, p. 224).
16
i.i.d.
I.e., they behave as the order statistics Y(1) , . . . , Y(n) , associated to Yi ∼ Uniform(0, t).
138
•
Proposition 3.118 — Merging independent Poisson processes
Let {N1 (t), t ≥ 0} and {N2 (t), t ≥ 0} be two independent Poisson processes with rates
λ1 and λ2 , respectively. Then the merged process {N1 (t) + N2 (t), t ≥ 0} is a Poisson
process with rate λ1 + λ2 .
•
Exercise 3.119 — Merging independent Poisson processes
Prove Proposition 3.118.
•
Exercise 3.120 — Merging independent Poisson processes
Men and women enter a supermarket according to independent Poisson processes having
respective rates two and four per minute.
(a) Starting at an arbitrary time, compute the probability that a least two men arrive
before three women arrive (Ross, 1989, p. 242, Exercise 20).
(b) What is the probability that the number of arrivals (men and women) exceeds ten
in the first 20 minutes?
•
Proposition 3.121 — Splitting a Poisson process (or sampling a Poisson
process)
Let {N (t), t ≥ 0} be a Poisson process with rate λ. Splitting the original Poisson process
based on a selection probability p yields two independent Poisson processes with rates
λp and λ(1 − p).
Moreover, we can add that N1 (t)|{N (t) = n} ∼ Binomial(n, p) and N2 (t)|{N (t) = n} ∼
Binomial(n, 1 − p).
•
139
Exercise 3.122 — Splitting a Poisson process
Prove Proposition 3.121 (Ross, 1989, pp. 218–219).
Why are the two resulting processes independent?
•
Exercise 3.123 — Splitting a Poisson process
If immigrants to area A arrive at a Poisson rate of ten per week, and if each immigrant
1
, then what is the probability that no people of
is of English descent with probability 12
English descent will emigrate to area A during the month of February (Ross, 1989, p.
220).
•
Exercise 3.124 — Splitting a Poisson process (Ross, 1989, p. 243, Exercise 23)
Cars pass a point on the highway at a Poisson rate of one per minute. If five percent of
the cars on the road are Dodges, then:
(a) What is the probability that at least one Dodge passes during an hour?
(b) If 50 cars have passed by an hour, what is the probability that five of them were
Dodges?
•
140
3.8
Generalizations of the Poisson process
In this section we consider three generalizations of the Poisson process. The first of these
is the non homogeneous Poisson process, which is obtained by allowing the arrival rate at
time t to be a function of t.
Definition 3.125 — Non homogeneous Poisson process (Ross, 1989, p. 234)
The counting process {N (t), t ≥ 0} is said to be a non homogeneous Poisson process with
intensity function λ(t) (t ≥ 0) if
• N (0) = 0;
• {N (t), t ≥ 0} has independent increments;
• P ({N (t + h) − N (t) = 1} = λ(t) × h + o(h), t ≥ 0;
• P ({N (t + h) − N (t) ≥ 2} = o(h), t ≥ 0.
Moreover,
Z
N (t + s) − N (s) ∼ Poisson
t+s
λ(z) dz
(3.44)
s
for s ≥ 0 and t > 0.
•
Exercise 3.126 — Non homogeneous Poisson process (“Stochastic Processes” test,
2003-01-14)
The number of arrivals to a shop is governed by a Poisson process with time dependent
rate
(
4 + 2t, 0 ≤ t ≤ 4
λ(t) =
24 − 3t, 4 < t ≤ 8.
(a) Obtain the expression of the expected value of the number of arrivals until t
(0 ≤ t ≤ 8). Derive the probability of no arrivals in the interval [3,5].
(b) Determine the expected values of the number of arrivals in the last 5 opening hours
(interval [3, 8]) given that 15 customers have arrived in the last 3 opening hours
(interval [5, 8]).
•
141
Exercise 3.127 — The output process of an infinite server Poisson queue and
the non homogeneous Poisson process
Prove that the output process of the M/G/∞ queue — i.e., the number of customers who
(by time t) have already left the infinite server queue with Poisson arrivals and general
service d.f. G — is a non homogeneous Poisson process with intensity function λG(t). •
Definition 3.128 — Compound Poisson process (Ross, 1989, p. 237)
A stochastic process {X(t), t ≥ 0} is said to be a compound Poisson process if it can be
represented as
X(t) =
N (t)
X
Yi ,
(3.45)
i=1
where
• {N (t), t ≥ 0} is a Poisson process with rate λ (λ > 0) and
i.i.d.
• Yi ∼ Y and independent of {N (t), t ≥ 0}.
•
Proposition 3.129 — Compound Poisson process (Ross, 1989, pp. 238–239)
Let {X(t), t ≥ 0} be a compound Poisson process. Then
E[X(t)] = λt × E[Y ]
(3.46)
V [X(t)] = λt × E[Y 2 ].
(3.47)
•
Exercise 3.130 — Compound Poisson process
Prove Proposition 3.129 by noting that E[X(t)] = E{E[X(t)|N (t)]} and
V [X(t)] = E{V [X(t)|N (t)]} + V {E[X(t)|N (t)]} (Ross, 1989, pp. 238–239).
•
Exercise 3.131 — Compound Poisson process (Ross, 1989, p. 239)
Suppose that families migrate to an area at a Poisson rate λ = 2 per week. Assume that
the number of people in each family is independent and takes values 1, 2, 3 and 4 with
respective probabilities 61 , 13 , 13 and 16 .
What is the expected value and variance of the number of individuals migrating to
this area during a five-week period?
•
142
Definition 3.132 — Conditional Poisson process (Ross, 1983, pp. 49–50)
Let:
• Λ be a positive r.v. having d.f. G; and
• {N (t), t ≥ 0} be a counting process such that, given that {Λ = λ}, {N (t), t ≥ 0}
is a Poisson process with rate λ.
Then {N (t), t ≥ 0} is called a conditional Poisson process and
Z +∞ −λt
e (λt)n
dG(λ).
P ({N (t + s) − N (s) = n}) =
n!
0
(3.48)
•
Remark 3.133 — Conditional Poisson process (Ross, 1983, p. 50)
{N (t), t ≥ 0} is not a Poisson process. For instance, whereas it has stationary increments,
it has not independent increments.
•
Exercise 3.134 — Conditional Poisson process
Suppose that, depending on factors not at present understood, the rate at which seismic
shocks occur in a certain region over a given season is either λ1 or λ2 . Suppose also that
the rate equals λ1 for p × 100% of the seasons and λ2 in the remaining time.
A simple model would be to suppose that {N (t), t ≥ 0} is a conditional Poisson
process such that Λ is either λ1 or λ2 with respective probabilities p and 1 − p.
Prove that the probability that it is a λ1 −season, given n shocks in the first t units of
a season, equals
p e−λ1 t (λ1 t)n
,
p e−λ1 t (λ1 t)n + (1 − p) e−λ2 t (λ2 t)n
(3.49)
by applying the Bayes’ theorem (Ross, 1983, p. 50).
143
•
References
• Barlow, R.E. and Proschan, F. (1965/1996). Mathematical Theory of Reliability.
SIAM (Classics in Applied Mathematics).
(TA169.BAR.64915)
• Barlow, R.E. and Proschan, F. (1975). Reliability and Life Testing. Holt, Rinehart
and Winston, Inc.
• Grimmett, G.R. and Stirzaker, D.R. (2001). Probability and Random Processes
(3rd. edition). Oxford University Press. (QA274.12-.76.GRI.30385 and QA274.12.76.GRI.40695 refer to the library code of the 1st. and 2nd. editions from 1982 and
1992, respectively.)
• Karr, A.F. (1993). Probability. Springer-Verlag.
• Pinkerton, S.D. and Holtgrave, D.R. (1998). The Bernoulli-process model in HIV
transmission: applications and implications. In Handbook of economic evaluation
of HIV prevention programs, Holtgrave, D.R. (Ed.), pp. 13–32. Plenum Press, New
York.
• Resnick, S.I. (1999). A Probability Path. Birkhäuser. (QA273.4-.67.RES.49925)
• Ross, S.M. (1983). Stochastic Processes. John Wiley & Sons.
.76.ROS.36921 and QA274.12-.76.ROS.37578)
(QA274.12-
• Ross, S.M. (1989). Introduction to Probability Models (fourth edition). Academic
Press. (QA274.12-.76.ROS.43540 refers to the library code of the 5th. revised edition
from 1993.)
144
Chapter 4
Expectation
One of the most fundamental concepts of probability theory and mathematical statistics
is the expectation of a r.v. (Resnick, 1999, p. 117).
Motivation 4.1 — Expectation (Karr, 1993, p. 101)
The expectation represents the center of gravity of a r.v. and has a measure theory
counterpart in integration theory.
Key computational formulas — not definitions of expectation — to obtain the
expectation of
• a discrete r.v. X with values in a countable set C and p.f. P ({X = x}) and
• the one of an absolutely continuous r.v. Y with p.d.f. fY (y)
are
E(X) =
X
x × P ({X = x})
(4.1)
x∈C
Z +∞
y × fY (y) dy,
E(Y ) =
(4.2)
−∞
respectively.
When X ≥ 0 it is permissible that E(X) = +∞, but finiteness is mandatory when X
can take both positive and negative (or null) values.
•
Remark 4.2 — Desired properties of expectation (Karr, 1993, p. 101)
1. Constant preserved
If X ≡ c then E(X) = c.
145
2. Monotonicity
If X ≤ Y
1
then E(X) ≤ E(Y ).
3. Linearity
For a, b ∈ IR, E(aX + bY ) = aE(X) + bE(Y ).
4. Continuity2
If Xn → X then E(Xn ) → E(X).
5. Relation to the probability
For each event A, E(1A ) = P (A).3
•
Expectation is to r.v. as probability is to events so that properties of expectation
extend those of probability.
4.1
Definition and fundamental properties
Many integration results are proved by first showing that they hold true for simple r.v.
and then extending the result to more general r.v. (Resnick, 1999, p. 117).
4.1.1
Simple r.v.
Let (Ω, F, P ) be a probability space and let us remind the reader that X is said to be a
simple r.v. if it assumes only finitely many values in which case
X=
n
X
ai × 1Ai ,
(4.3)
i=1
where:
• a1 , . . . , an are real numbers not necessarily distinct;
• {A1 , . . . , An } constitutes a partition of Ω;
• 1Ai is the indicator function of event Ai , i = 1, . . . , n.
1
I.e. X(ω) ≤ Y (ω), ∀ ω ∈ Ω.
Continuity is not valid without restriction.
3
Recall that 1Ai (ω) = 1, if w ∈ Ai , and 1Ai (ω) = 0, otherwise.
2
146
Definition 4.3 — Expectation of a simple r.v. (Karr, 1993, p. 102)
P
The expectation of the simple r.v. X = ni=1 ai × 1Ai is given by
E(X) =
n
X
ai × P (Ai ).
(4.4)
i=1
•
Remark 4.4 — Expectation of a simple r.v. (Resnick, 1999, p. 119; Karr, 1993, p.
102)
• Note that Definition 4.3 coincides with our knowledge of discrete probability from
more elementary courses: the expectation is computed by taking a possible value,
multiplying by the probability of the possible value and then summing over all
possible values.
• E(X) is well-defined in the sense that all representations of X yield the same value
P
P
a0 ×1A0j ,
for E(X): different representations of X, X = ni=1 ai ×1Ai and X = m
Pm 0 j=1 j0
Pn
lead to the same expected value E(X) = i=1 ai × P (Ai ) = j=1 aj × P (Aj ).
• The expectation of an indicator function is indeed the probability of the associated
event.
•
Proposition 4.5 — Properties of the set of simple r.v. (Resnick, 1999, p. 118)
Let E be the set of all simple r.v. defined on (Ω, F, P ). We have the following properties
of E.
1. E is a vector space, i.e.:
P
ai × 1Ai ∈ E and α ∈ IR then αX = ni=1 αai × 1Ai ∈ E; and
P
P
(b) If X = ni=1 ai × 1Ai ∈ E and Y = m
j=1 bj × 1Bj ∈ E then
(a) if X =
Pn
i=1
X +Y =
m
n X
X
(ai + bj ) × 1Ai ∩Bj ∈ E.
(4.5)
i=1 j=1
2. If X, Y ∈ E then XY ∈ E since
XY =
n X
m
X
(ai × bj ) × 1Ai ∩Bj .
(4.6)
i=1 j=1
•
147
Proposition 4.6 — Expectation of a linear combination of simple r.v. (Karr,
1993, p. 103)
Let X and Y be two simple r.v. and a, b ∈ IR. Then aX + bY is also a simple r.v. and
E(aX + bY ) = aE(X) + bE(Y ).
(4.7)
•
Exercise 4.7 — Expectation of a linear combination of simple r.v.
Prove Proposition 4.6 by capitalizing on Proposition 4.5 (Karr, 1993, p. 103).
•
Exercise 4.8 — Expectation of a sum of discrete r.v. in a distribution problem
(Walrand, 2004, p. 51, Example 4.10.6)
Suppose you put m balls randomly in n boxes. Each box can hold an arbitrarily large
number of balls.
m
Prove that the expected number of empty boxes is equal to n × n−1
.
•
n
Exercise 4.9 — Expectation of a sum of discrete r.v. in a selection problem
(Walrand, 2004, p. 52, Example 4.10.7)
A cereal company is running a promotion for which it is giving a toy in every box of
cereal. There are n different toys and each box is equally likely to contain any one of the
n toys.
Prove that the expected number of boxes of cereal you have to purchase to collect all
P
n toys is given by n × nm=1 m1 .
•
Remark 4.10 — Monotonicity of expectation for simple r.v. (Karr, 1993, p. 103)
The monotonicity of expectation for simple r.v. is a desired property which follows from
• linearity and
• positivity (or better said non negativity) — if X ≥ 0 then E(X) ≥ 0 —, a seemingly
weaker property of expectation.
In fact, if X ≤ Y ⇔ Y − X ≥ 0 then
• E(Y ) − E(X) = E(Y − X) ≥ 0.
This argument is valid provided that E(Y ) − E(X) is not of the form +∞ − ∞.
148
•
Proposition 4.11 — Monotonicity of expectation for simple r.v. (Karr, 1993, p.
103)
Let X and Y be two simple r.v. such that X ≤ Y . Then E(X) ≤ E(Y ).
•
Example 4.12 — On the (dis)continuity of expectation of simple r.v. (Karr,
1993, pp. 103–104)
Continuity of expectation fails even for simple r.v. Let P be the uniform distribution on
[0, 1] and
Xn = n × 1(0, 1 ) .
(4.8)
n
Then Xn (ω) → 0, ∀ w ∈ Ω, but E(Xn ) = 1, for each n.
Thus, we need additional conditions to guarantee continuity of expectation.
4.1.2
•
Non negative r.v.
Before we proceed with the definition of the expectation of non negative r.v.,4 we need
to recall the measurability theorem. This theorem state that any non negative r.v. can
be approximated by a simple r.v., and it is the reason why it is often the case that an
integration result about non negative r.v. — such as the expectation and its properties
— is proven first to simple r.v.
Theorem 4.13 — Measurability theorem (Resnick, 1999, p. 91; Karr, 1993, p. 50)
Suppose X(ω) ≥ 0, for all ω. Then X : Ω → IR is a Borel measurable function (i.e.
a r.v.) iff there is an increasing sequence of simple and non negative r.v. X1 , X2 , . . .
(0 ≤ X1 ≤ X2 ≤ . . .) such that
Xn ↑ X,
(4.9)
(Xn (ω) ↑ X(ω), for every ω).
•
Exercise 4.14 — Measurability theorem
Prove Theorem 4.13 by considering
n
Xn =
n2
X
k−1
k=1
2n
1{ k−1
+ n × 1{X≥n} ,
k
n ≤X< n }
2
for each n (Resnick, 1999, p. 118; Karr, 1993, p. 50).
4
(4.10)
2
Karr (1993) and Resnick (1999) call these r.v. positive when they are actually non negative.
149
•
Motivation 4.15 — Expectation of a non negative r.v. (Karr, 1993, pp. 103–104)
We now extend the definition of expectation to all non negative r.v. However, we have
already seen that continuity of expectation fails even for simple r.v. and therefore we
cannot define the expected value of a non negative r.v. simply as E(X) = limn→+∞ E(Xn ).
Unsurprisingly, if we apply the measurability theorem then the definition of
expectation of a non negative r.v. virtually forces monotone continuity for increasing
sequences of non negative r.v.:
• if X1 , X2 , . . . are simple and non negative r.v. and X is a non negative r.v. such that
Xn ↑ X (pointwise) then E(Xn ) ↑ E(X).
It is convenient and useful to assume that these non negative r.v. can take values in the
+
extended set of non negative real numbers, IR0 .
Further on, we shall have to establish another restricted form of continuity: dominated
continuity for integrable r.v.5
•
Definition 4.16 — Expectation of a non negative r.v. (Karr, 1993, p. 104)
The expectation of a non negative r.v. X is
E(X) = lim E(Xn ) ≤ +∞,
(4.11)
n→+∞
where Xn are simple and non negative r.v. such that Xn ↑ X.
The expectation of X over the event A is E(X; A) = E(X × 1A ).
•
Remark 4.17 — Expectation of a non negative r.v. (Karr, 1993, p. 104)
The limit defining E(X)
+
• exists in the set of extended non negative real numbers IR0 , and
• does not depend on the approximating sequence {Xn , n ∈ IN }, as stated in the next
proposition.
•
Proposition 4.18 — Expectation of a non negative r.v. (Karr, 1993, p. 104)
Let {Xn , n ∈ IN } and {X̃m , m ∈ IN } be sequences of simple and non negative r.v.
increasing to X. Then
lim E(Xn ) = lim E(X̃m ).
n→+∞
(4.12)
m→+∞
•
5
We shall soon define integrable r.v.
150
Exercise 4.19 — Expectation of a non negative r.v.
Prove Proposition 4.18 (Karr, 1993, p. 104; Resnick, 1999, pp. 122–123).
•
We now list some basic properties of the expectation operator applied to non negative
r.v. For instance, linearity, monotonicity and monotone continuity/convergence. This last
property describes how expectation and limits interact, and under which circunstances
we are allowed to interchange expectation and limits.
Proposition 4.20 — Expectation of a linear combination of non negative r.v.
(Karr, 1993, p. 104; Resnick, 1999, p. 123)
Let X and Y be two non negative r.v. and a, b ∈ IR+ . Then
E(aX + bY ) = aE(X) + bE(Y ).
(4.13)
•
Exercise 4.21 — Expectation of a linear combination of non negative r.v.
Prove Proposition 4.20 by considering two sequences of simple and non negative r.v.
{Xn , n ∈ IN } and {Yn , n ∈ IN } such that Xn ↑ X and Yn ↑ Y — and, thus, (aXn + bYn ) ↑
(aX + bY ) — (Karr, 1993, p. 104).
•
Corollary 4.22 — Monotonicity of expectation for non negative r.v. (Karr,
1993, p. 105)
Let X and Y be two non negative r.v. such that X ≤ Y . Then E(X) ≤ E(Y ).
•
Remark 4.23 — Monotonicity of expectation for non negative r.v. (Karr, 1993,
p. 105)
Monotonicity of expectation follows, once again, from positivity and linearity.
•
Motivation 4.24 — Fatou’s lemma (Karr, 1993, p. 105)
The next result plays a vital role in the definition of monotone continuity/convergence. •
Theorem 4.25 — Fatou’s lemma (Karr, 1993, p. 105; Resnick, 1999, p. 132)
Let {Xn , n ∈ IN } be a sequence of non negative r.v. Then
E(lim inf Xn ) ≤ lim inf E(Xn ).
(4.14)
•
151
Remark 4.26 — Fatou’s lemma (Karr, 1993, p. 105)
The inequality (4.14) in Fatou’s lemma can be strict. For instance, in Example 4.12 we
are dealing with E(lim inf Xn ) = 0 < lim inf E(Xn ) = 1.
•
Exercise 4.27 — Fatou’s lemma
Prove Theorem 4.25 (Karr, 1993, p. 105; Resnick, 1999, p. 132).
•
Exercise 4.28 — Fatou’s lemma and continuity of p.f.
Verify that Theorem 4.25 could be used in a part of the proof of the continuity of p.f. if
we considered Xn = 1An (Karr, 1993, p. 106).
•
We now state another property of expectation of non negative r.v.: the monotone
continuity/convergence of expectation.
Theorem 4.29 — Monotone convergence theorem (Karr, 1993, p. 106; Resnick,
1999, pp. 123–124)
Let {Xn , n ∈ IN } be an increasing sequence of non negative r.v. and X a non negative
r.v. If
Xn ↑ X
(4.15)
E(Xn ) ↑ E(X).
(4.16)
then
•
Remark 4.30 — Monotone convergence theorem (Karr, 1993, p. 106)
The sequence of simple and non negative r.v. from Example 4.12, Xn = n × 1(0, 1 ) , does
n
not violate the monotone convergence theorem because in that instance it is not true that
Xn ↑ X.6
•
Exercise 4.31 — Monotone convergence theorem
Prove Theorem 4.29 (Karr, 1993, p. 106; Resnick, 1999, pp. 124–125, for a more
sophisticated proof).
•
Please note that the sequence is not even increasing: n increases but the sequence sets (0, n1 ) is a
decreasing one.
6
152
Exercise 4.32 — Monotone convergence theorem and monotone continuity of
p.f.
Verify that Theorem 4.29 could be used to prove the monotone continuity of p.f. if we
considered Xn = 1An and X = 1A , where An ↑ A (Karr, 1993, p. 106).
•
One of the implication of the monotone convergence theorem is the linearity of
expectation for convergent series, and is what Resnick (1999, p. 131) calls the series version
of the monotone convergence theorem. This results refers under which circunstances we
are allowed to interchange expectation and limits.
Theorem 4.33 — Expectation of a linear convergent series of non negative r.v.
(Karr, 1993, p. 106; Resnick, 1999, p. 131)
P
Let {Yk , k ∈ IN } be a collection of non negative r.v. such that +∞
k=1 Yk (ω) < +∞, for
every ω. Then
! +∞
+∞
X
X
E
Yk =
E(Yk ).
(4.17)
k=1
k=1
•
Exercise 4.34 — Expectation of a linear convergent series of non negative r.v.
Pn
Prove Theorem 4.33 by considering Xn =
k=1 Yk and applying the monotone
convergence theorem (Karr, 1993, p. 106).
•
Exercise 4.35 — Expectation of a linear convergent series of non negative r.v.
and σ−additivity
Verify that Theorem 4.33 could be used to prove the σ−additivity of p.f. if we considered
P
S
•
Yk = 1Ak , where Ak are disjoint, so that +∞
k=1 Yk = 1 +∞ Ak (Karr, 1993, p. 106).
k=1
Proposition 4.36 – “Converse” of the positivity of expectation (Karr, 1993, p.
107)
a.s.
Let X be a non negative r.v. If E(X) = 0 then X = 0.
•
Exercise 4.37 – “Converse” of the positivity of expectation
Prove Proposition 4.36 (Karr, 1993, p. 107).
153
•
4.1.3
Integrable r.v.
It is time to extend the definition of expectation to r.v. X that can take both positive
and negative (or null) values. But first recall that:
• X + = max{X, 0} represents the positive part of the r.v. X;
• X − = − min{X, 0} = max{ − X, 0} represents the negative part of the r.v. X;
• X = X + − X −;
• |X| = X + + X − .
The definition of expectation of such a r.v. preserves linearity and is based on the fact
that X can be written as a linear combination of two non negative r.v.: X = X + − X − .
Definition 4.38 — Integrable r.v.; the set of integrable r.v. (Karr, 1993, p. 107;
Resnick, 1999, p. 126)
Let X be a r.v., not necessarily non negative. Then X is said to be integrable if
E(|X|) < +∞.
The set of integrable r.v. is denoted by L1 or L1 (P ) if the probability measure needs
to be emphasized.
•
Definition 4.39 — Expectation of an integrable r.v. (Karr, 1993, p. 107)
Let X be an integrable r.v. Then the expectation of X is given by
E(X) = E(X + ) − E(X − ).
(4.18)
For an event A, the expectation of X over A is E(X; A) = E(X × 1A ).
•
Remark 4.40 — Expectation of an integrable r.v. (Karr, 1993, p. 107; Resnick,
1999, p. 126)
1. If X is an integrable r.v. then
E(X + ) + E(X − ) = E(X + + X − ) = E(|X|) < +∞
(4.19)
so both E(X + ) and E(X − ) are finite, E(X + ) − E(X − ) is not of the form ∞ − ∞,
thus, the definition of expectation of X is coherent.
154
2. Moreover, since |X × 1A | ≤ |X|, E(X; A) = E(X × 1A ) is finite (i.e. exists!) as long
as E(|X|) < +∞, that is, as long as E(X) exists.
3. Some conventions when X is not integrable...
If E(X + ) < +∞ but E(X − ) = +∞ then we consider E(X) = −∞.
If E(X + ) = +∞ but E(X − ) < +∞ then we take E(X) = +∞.
If E(X + ) = +∞ and E(X − ) = +∞ then E(X) does not exist.
•
What follows refers to properties of the expectation operator.
Theorem 4.41 — Expectation of a linear combination of integrable r.v. (Karr,
1993, p. 107)
Let X and Y be two integrable r.v. — i.e., X, Y ∈ L1 — and a, b ∈ IR. Then aX + bY is
also an integrable r.v.7 and
E(aX + bY ) = aE(X) + bE(Y ).
(4.20)
•
Exercise 4.42 — Expectation of a linear combination of integrable r.v.
Prove Theorem 4.41 (Karr, 1993, p. 108).
•
Corollary 4.43 — Modulus inequality (Karr, 1993, p. 108; Resnick, 1999, p. 128)
If X ∈ L1 then
|E(X)| ≤ E(|X|).
(4.21)
•
Exercise 4.44 — Modulus inequality
Prove Corollary 4.43 (Karr, 1993, p. 108).
•
Corollary 4.45 — Monotonicity of expectation for integrable r.v. (Karr, 1993,
p. 108; Resnick, 1999, p. 127)
If X, Y ∈ L1 and X ≤ Y then
E(X) ≤ E(Y ).
(4.22)
•
Exercise 4.46 — Monotonicity of expectation for integrable r.v.
Prove Corollary 4.45 (Resnick, 1999, pp. 127–128).
7
That is, aX + bY ∈ L1 . In fact, L1 is a vector space.
155
•
The continuity of expectation for integrable r.v. can be finally stated.
Theorem 4.47 — Dominated convergence theorem (Karr, 1993, p. 108; Resnick,
1999, p. 133)
Let X1 , X2 , . . . ∈ L1 and X ∈ L1 with
Xn → X.
(4.23)
If there is a dominating r.v. Y ∈ L1 such that
|Xn | ≤ Y,
(4.24)
for each n, then
lim E(Xn ) = E(X).
(4.25)
n→+∞
•
Remark 4.48 — Dominated convergence theorem (Karr, 1993, p. 109)
The sequence of simple r.v., Xn = n × 1(0, 1 ) , from Example 4.12 does not violate the
n
dominated convergence theorem because any r.v. Y dominating Xn for each n must satisfy
P
1
1
1 , which implies that E(Y ) = +∞, thus Y 6∈ L
Y ≥ +∞
and we cannot
,n
)
n=1 n × 1( n+1
apply Theorem 4.47.
•
Exercise 4.49 — Dominated convergence theorem
Prove Theorem 4.47 (Karr, 1993, p. 109; Resnick, 1999, p. 133, for a detailed proof).
•
Exercise 4.50 — Dominated convergence theorem and continuity of p.f.
Verify that Theorem 4.47 could be used to prove the continuity of p.f. if we considered
Xn = 1An , where An → A, and Y ≡ 1 as the dominating integrable r.v. (Karr, 1993, p.
109).
•
4.1.4
Complex r.v.
Definition 4.51 — Integrable complex r.v.; expectation of a complex r.v. (Karr,
1993, p. 109)
√
A complex r.v. Z = X + iY ∈ L1 if E(|Z|) = E( X 2 + Y 2 ) < +∞, and in this case the
expectation Z is E(Z) = E(X) + iE(Y ).
•
156
4.2
Integrals with respect to distribution functions
Integrals (of Borel measurable functions) with respect to d.f. are known as Lebesgue–
Stieltjes integrals. Moreover, they are really expectations with respect to probabilities on
IR and are reduced to sums and Riemann (more generally, Lebesgue) integrals.
4.2.1
On integration
Remark 4.52 — Riemann integral (http://en.wikipedia.org/wiki/Riemann integral)
In the branch of mathematics known as real analysis, the Riemann integral, created
by Bernhard Riemann (1826–1866), was the first rigorous definition of the integral of a
function on an interval.
• Overview
Let g be a non-negative real-valued function of the interval [a, b], and let
S = {(x, y) : 0 < y < g(x)} be the region of the plane under the graph of the
function g and above the interval [a, b].
The basic idea of the Riemann integral is to use very simple approximations for the
Rb
area of S, denoted by a g(x) dx, namely by taking better and better approximations
— we can say that “in the limit” we get exactly the area of S under the curve.
• Riemann sums
Choose a real-valued function f which is defined on the interval [a, b]. The Riemann
sum of f with respect to the tagged partition a = x0 < x1 < x2 < . . . < xn = b
together with t0 , . . . , tn−1 (where xi ≤ ti ≤ xi+1 ) is
n−1
X
g(ti )(xi+1 − xi ),
(4.26)
i=0
where each term represents the area of a rectangle with height g(ti ) and length
xi+1 − xi . Thus, the Riemann sum is the signed area under all the rectangles.
• Riemann integral
Loosely speaking, the Riemann integral is the limit of the Riemann sums of a
function as the partitions get finer.
If the limit exists then the function is said to be integrable (or more specifically
Riemann-integrable).
157
• Limitations of the Riemann integral
With the advent of Fourier series, many analytical problems involving integrals came
up whose satisfactory solution required interchanging limit processes and integral
signs.
Failure of monotone convergence — The indicator function 1Q on the rationals is not
Riemann integrable. No matter how the set [0, 1] is partitioned into subintervals,
each partition will contain at least one rational and at least one irrational number,
since rationals and irrationals are both dense in the reals. Thus, the upper Darboux
sums8 will all be one, and the lower Darboux sums9 will all be zero.
Unsuitability for unbounded intervals — The Riemann integral can only integrate
functions on a bounded interval. It can however be extended to unbounded intervals
by taking limits, so long as this does not yield an answer such as +∞ − ∞.
What about integrating on structures other than Euclidean space? — The Riemann
integral is inextricably linked to the order structure of the line. How do we free
ourselves of this limitation?
•
Remark 4.53 — Lebesgue integral (http://en.wikipedia.org/wiki/Lebesgue integral;
http://en.wikipedia.org/wiki/Henri Lebesgue)
Lebesgue integration plays an important role in real analysis, the axiomatic theory of
probability, and many other fields in the mathematical sciences. The Lebesgue integral is
a construction that extends the integral to a larger class of functions defined over spaces
more general than the real line.
• Lebesgue’s theory of integration
Henri Léon Lebesgue (1875–1941) invented a new method of integration to solve
this problem. Instead of using the areas of rectangles, which put the focus on the
domain of the function, Lebesgue looked at the codomain of the function for his
fundamental unit of area. Lebesgue’s idea was to first build the integral for what he
called simple functions, measurable functions that take only finitely many values.
Then he defined it for more complicated functions as the least upper bound of all
the integrals of simple functions smaller than the function in question.
Pn−1
The upper Darboux sum of g with respect to the partition is
i=0 (xi+1 − xi )Mi+1 , where
Mi+1 = supx∈[xi ,xi+1 ] g(x).
Pn
9
The lower Darboux sum of g with respect to the partition is
i=0 (xi+1 − xi )mi+1 , where
mi+1 = inf x∈[xi ,xi+1 ] g(x).
8
158
Lebesgue integration has the beautiful property that every bounded function defined
over a bounded interval with a Riemann integral also has a Lebesgue integral, and
for those functions the two integrals agree. But there are many functions with a
Lebesgue integral that have no Riemann integral.
As part of the development of Lebesgue integration, Lebesgue invented the concept
of Lebesgue measure, which extends the idea of length from intervals to a very large
class of sets, called measurable sets.
• Integration
We start with a measure space (Ω, F, µ) where Ω is a set, F is a σ − algebra of
subsets of Ω and µ is a (non-negative) measure on F of subsets of Ω.
In the mathematical theory of probability, we confine our study to a probability
measure µ, which satisfies µ(Ω) = 1.
In Lebesgue’s theory, integrals are defined for a class of functions called measurable
functions.
R
We build up an integral Ω g dµ for measurable real-valued functions g defined on Ω
in stages:
– Indicator functions. To assign a value to the integral of the indicator
function of a measurable set S consistent with the given measure µ, the only
reasonable choice is to set
Z
1S dµ = µ(S).
Ω
– Simple functions. A finite linear combination of indicator functions
P
k ak 1Sk . When the coefficients ak are non-negative, we set
Z X
X Z
X
ak 1Sk ) dµ =
ak
ak µ(Sk ).
(
1Sk dµ =
Ω
k
Ω
k
k
– Non-negative functions. We define
Z
Z
g dµ = sup
s dµ : 0 ≤ s ≤ g, s simple .
Ω
Ω
– Signed functions. g = g + − g − ... And it makes sense to define
Z
Z
Z
+
gdµ =
g dµ − g − dµ.
Ω
Ω
Ω
•
159
Remark 4.54 — Lebesgue/Riemann–Stieltjes integration
(http://en.wikipedia.org/wiki/Lebesgue-Stieltjes integration)
The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure
known as the Lebesgue–Stieltjes measure, which may be associated to any function of
bounded variation on the real line.
• Definition
Rb
The Lebesgue–Stieltjes integral a g(x) dF (x) is defined when g : [a, b] → IR is Borelmeasurable and bounded and F : [a, b] → IR is of bounded variation in [a, b] and
right-continuous, or when g is non-negative and F is monotone and right-continuous.
• Riemann–Stieltjes integration and probability theory
When g is a continuous real-valued function of a real variable and F is a
non-decreasing real function, the Lebesgue–Stieltjes integral is equivalent to the
Rb
Riemann–Stieltjes integral, in which case we often write a g(x) dF (x) for the
Lebesgue–Stieltjes integral, letting the measure PF remain implicit.
This is particularly common in probability theory when F is the cumulative
distribution function of a real-valued random variable X, in which case
Z ∞
g(x) dF (x) = EF [g(X)].
−∞
•
4.2.2
Generalities
First of all, we should recall that given a d.f. F on IR, there is a unique p.f. on IR such
that PF ((a, b]) = F (b) − F (a).
Moreover, all functions g appearing below are assumed to be Borel measurable.
Definition 4.55 — Integral of a nonnegative g with respect to a d.f. (Karr, 1993,
p. 110)
Let F be a d.f. on IR and g a non negative function. Then the integral of g with respect
to F is given by
Z
g(x) dF (x) = EF (g) ≤ +∞,
(4.27)
IR
where the expectation is that of g(X) as a Borel measurable function of the r.v. X defined
on the probability space (IR, B(IR), PF ).
•
160
Definition 4.56 — Integrable of a function with respect to a d.f. (Karr, 1993, p.
110)
Let F be a d.f. on IR and g a signed function. Then g is said to be integrable with respect
R
to F if IR g(x) dF (x) < +∞, and this case, the integral of g with respect to F equals
Z
Z
Z
+
g(x) dF (x) =
g (x) dF (x) −
g − (x) dF (x).
(4.28)
IR
IR
IR
•
Definition 4.57 — Integral of a function over a set with respect to a d.f. (Karr,
1993, p. 110)
Let F be a d.f. on IR and g either non negative or integrable and B ∈ B(IR). The integral
of g over B with respect to F is equal to
Z
Z
g(x) × 1B (x) dF (x).
(4.29)
g(x) dF (x) =
B
IR
•
The properties of the integral of a function with respect to a d.f. are those of
expectation:
1. Constant preserved
2. Monotonicity
3. Linearity
4. Relation to PF
5. Fatou’s lemma
6. Monotone convergence theorem
7. Dominated convergence theorem
161
4.2.3
Discrete distribution functions
Keep in mind that integrals with respect to discrete d.f. are sums.
Theorem 4.58 — Integral with respect to a discrete d.f. (Karr, 1993, p. 111)
Consider a d.f. F (t) that can be written as
X
F (x) =
pi × 1[xi ,+∞) (x).
(4.30)
i
Then, for each g ≥ 0,
Z
X
g(x) dF (x) =
g(xi ) × pi .
IR
(4.31)
i
•
Exercise 4.59 — Integral with respect to a discrete d.f.
Prove Theorem 4.58 (Karr, 1993, p. 111).
•
Corollary 4.60 — Integrable function with respect to a discrete d.f. (Karr, 1993,
p. 111)
The function g is said to be integrable with respect to the discrete d.f. F iff
X
|g(xi )| × pi < +∞,
(4.32)
i
and in this case
Z
X
g(x) dF (x) =
g(xi ) × pi .
IR
4.2.4
(4.33)
i
•
Absolutely continuous distribution functions
Now note that integrals with respect to absolutely continuous d.f. are Riemann integrals.
Theorem 4.61 — Integral with respect to an absolutely continuous d.f. (Karr,
1993, p. 112)
Suppose that the d.f. F is absolutely continuous and is associated to a piecewise continuous
p.d.f. f . If g is a non negative function and piecewise continuous then
Z
Z +∞
g(x) dF (x) =
g(x) × f (x) dx,
(4.34)
IR
−∞
where the integral on the right-hand side is an improper Riemann integral.
162
•
Theorem 4.62 — Integral with respect to an absolutely continuous d.f.
Prove Theorem 4.61 (Karr, 1993, p. 112).
•
Corollary 4.63 — Integral with respect to an absolutely continuous d.f. (Karr,
1993, p. 112)
A piecewise continuous function g is said to be integrable with respect to the d.f. F iff
Z +∞
|g(x)|f (x) dx < +∞,
(4.35)
−∞
and in this case
Z
Z
g(x) dF (x) =
g(x) × f (x) dx.
(4.36)
−∞
IR
4.2.5
+∞
•
Mixed distribution functions
Recall that a mixed d.f. F is a convex combination of a discrete d.f.
X
Fd (x) =
pi × 1(xi ≤ x)
(4.37)
i
and an absolutely continuous d.f.
Z x
fa (s) ds.
Fa (x) =
(4.38)
−∞
Thus,
F (x) = α × Fd (x) + (1 − α) × Fa (x),
(4.39)
where α ∈ (0, 1).
Corollary 4.64 — Integral with respect to a mixed d.f. (Karr, 1993, p. 112)
The integral of g with respect to the mixed d.f. F is a corresponding combination of
integrals with respect to Fd and Fa :
Z
Z
Z
g(x) dF (x) = α ×
g(x) dFd (x) + (1 − α) ×
g(x) dFa (x)
IR
IR
IR
Z +∞
X
= α×
g(xi ) × pi + (1 − α) ×
g(x) × fa (x) dx.
(4.40)
−∞
i
In order that the integral with respect to a mixed d.f. exists, g must be piecewise
continuous and either non negative or integrable with respect to both Fd and Fa .
•
163
4.3
Computation of expectations
So far we have defined the expectation for simple r.v.
The expectations of other types of r.v. — such as non negative, integrable and mixed
r.v. — naturally involve integrals with respect to distribution functions.
4.3.1
Non negative r.v.
The second equality in the next formula is quite convenient because it allows us to obtain
the expectation of a non negative r.v. — be it a discrete, absolutely continuous or mixed
— in terms of an improper Riemann integral.
Theorem 4.65 — Expected value of a non negative r.v. (Karr, 1993, p. 113)
If X ≥ 0 then
Z +∞
Z +∞
E(X) =
x dFX (x) =
[1 − FX (x)] dx.
(4.41)
0
0
•
Exercise 4.66 — Expected value of a non negative r.v.
Prove Theorem 4.65 (Karr, 1993, pp. 113–114).
•
Corollary 4.67 — Expected value of a non negative integer-valued r.v. (Karr,
1993, p. 114)
Let X be a non negative integer-valued r.v. Then
E(X) =
+∞
X
n=1
n × P ({X = n}) =
+∞
X
P ({X ≥ n}) =
n=1
+∞
X
P ({X > n}).
n=0
Exercise 4.68 — Expected value of a non negative integer-valued r.v.
Prove Corollary 4.67 (Karr, 1993, p. 114).
(4.42)
•
•
Corollary 4.69 — Expected value of a non negative absolutely continuous r.v.
Let X be a non negative absolutely continuous r.v. with p.d.f. fX (x).
Z +∞
Z +∞
E(X) =
x × fX (x) dx =
[1 − FX (x)] dx.
(4.43)
0
0
•
164
Exercise 4.70 — A nonnegative r.v. with infinite expectation
Let X ∼ Pareto(b = 1, α = 1). i.e.
(
α bα
= x12 , x ≥ b = 1
xα+1
fX (x) =
0,
otherwise.
Prove that E(X) exists and E(X) = +∞ (Resnick, 1999, p. 126, Example 5.2.1).
4.3.2
(4.44)
•
Integrable r.v.
Let us remind the reader that X is said to be an integrable r.v. if E(|X|) = E(X + ) +
E(X − ) < +∞.
Theorem 4.71 — Expected value of an integrable r.v. (Karr, 1993, p. 114)
If X is an integrable r.v. then
Z +∞
x dFX (x).
(4.45)
E(X) =
−∞
•
Exercise 4.72 — Expected value of an integrable r.v.
Prove Theorem 4.71 (Karr, 1993, p. 114).
•
Corollary 4.73 — Expected value of an integrable absolutely continuous r.v.
Let X be an integrable absolutely continuous r.v. with p.d.f. fX (x) then
Z +∞
x × fX (x) dx.
(4.46)
E(X) =
−∞
•
Exercise 4.74 — Real r.v. without expectation
Prove that E(X + ) = E(X − ) = +∞ — and therefore E(X) does not exist — if X has
p.d.f. equal to:
(
1
, |x| > 1
2x2
(a)
fX (x) =
(4.47)
0,
otherwise;
(b)
fX (x) =
1
, x ∈ IR.
π(1 + x2 )
(4.48)
(Resnick, 1999, p. 126, Example 5.2.1.)10
•
10
There is a typo in the definition of the first p.d.f. in Resnick (1999): x > 1 should read as |x| > 1.
The second p.d.f. corresponds to the one of a r.v. with (standard) Cauchy distribution.
165
4.3.3
Mixed r.v.
When dealing with mixed r.v. X we take advantage of the fact that FX (x) is a convex
combination of the d.f. of a discrete r.v. Xd and the d.f. of an absolutely continuous r.v.
Xa .
Corollary 4.75 — Expectation of a mixed r.v.
The expected value of the mixed r.v. X with d.f. FX (x) = α × FXd (x) + (1 − α) × FXa (x),
where α ∈ (0, 1), is given by
Z
Z
Z
x dFX (x) = α ×
x dFXd (x) + (1 − α) ×
x dFXa (x)
IR
IR
IR
X
= α×
xi × P ({Xd = xi })
i
Z
+∞
+(1 − α) ×
x × fXa (x) dx.
(4.49)
−∞
•
Exercise 4.76 — Expectation of a mixed r.v.
A random variable X has the following d.f.:11

x<0

 0,

 0.3,
0≤x<2
FX (x) =

0.3 + 0.2x, 2 ≤ x < 3



1,
x ≥ 3.
(4.50)
(a) Why is X a mixed r.v.?
(b) Write FX (x) a linear combination of the d.f. of two r.v.: a discrete and an absolutely
continuous r.v.
(c) Obtain the expected value of X, by using the fact that X is non negative, thus,
R +∞
E(X) = 0 [1 − FX (x)] dx.
Compare this value with the one you would obtain using Corollary 4.75.
11
Adapted from Walrand (2004, pp. 53–55, Example 4.10.9).
166
•
Exercise 4.77 — Expectation of a mixed r.v. in a queueing setting
Consider a M/M/1 system.12 Let:
• Ls be the number of customers an arriving customer finds in the system in
equilibrium;13
• Wq be the waiting time in queue of this arriving customer.14
Under these conditions, we can state that
P (Ls = k) = (1 − ρ) × ρk , k ∈ IN0 ;
thus, Ls ∼ geometric∗ (1 − ρ), where ρ =
(4.51)
λ
µ
∈ (0, 1) and E(Ls ) =
ρ
.
1−ρ
(a) Argue that Wq |{Ls = k} ∼ Gamma(k, µ), for k ∈ IN .
(b) Prove that Wq |{Wq > 0} ∼ Exponential(µ(1 − ρ)).
(c) Demonstrate that Wq is a mixed r.v. with d.f. given by:
(
FWq (w) =
0,
w<0
(1 − ρ) + ρ × FExp(µ(1−ρ)) (w), w ≥ 0.
(d) Verify that E(Wq ) =
ρ
.
µ(1−ρ)
(4.52)
•
12
The arrivals to the system are governed by a Poisson process with rate λ, i.e. the time between arrivals
has an exponential distribution with parameter λ; needless to say, M stands for memoryless. The service
times are not only i.i.d. with exponential distribution with parameter µ, but also independent from the
arrival process. There is only one server, and the service policy is FCFS (first come first served). ρ = µλ
represents the traffic intensity and we assume that ρ ∈ (0, 1).
13
Equilibrium roughly means that a lot of time has elapsed since the system has been operating and
therefore the initial conditions no longer influence the state of system.
14
Wq is the time elapsed from the moment the customer arrives until his/her service starts in the
system in equilibrium.
167
4.3.4
Functions of r.v.
Unsurprisingly, we are surely able to derive expressions for the expectation of a Borel
measurable function g of the r.v. X, E[g(X)]. Obtaining this expectation does not require
the derivation of the d.f. of g(X) and follows from section 4.2.
In the two sections we shall discuss the expectation of specific functions of r.v.:
g(X) = X k , k ∈ IN .
Theorem 4.78 — Expected value of a function of a r.v. (Karr, 1993, p. 115)
Let X be a r.v., and g be a Borel measurable function either non negative or integrable.
Then
Z
E[g(X)] =
g(x) dFX (x).
(4.53)
IR
•
Exercise 4.79 — Expected value of a function of a r.v.
Prove Theorem 4.78 (Karr, 1993, p. 115).
•
Corollary 4.80 — Expected value of a function of a discrete r.v. (Karr, 1993, p.
115)
Let X be a discrete r.v., and g be a Borel measurable function either non negative or
integrable. Then
X
g(xi ) × P ({X = xi }).
(4.54)
E[g(X)] =
xi
•
Corollary 4.81 — Expected value of a function of an absolutely continuous r.v.
(Karr, 1993, p. 115)
Let X be an absolutely continuous r.v., and g be a Borel measurable function either non
negative or integrable. Then
Z +∞
E[g(X)] =
g(x) × fX (x) dx.
(4.55)
−∞
•
168
4.3.5
Functions of random vectors
When dealing with functions of random vectors, the only useful formulas are those
referring to the expectation of functions of discrete and absolutely continuous random
vectors.
These formulas will be used to obtain, for instance, what we shall call measures of
(linear) association between r.v.
Theorem 4.82 — Expectation of a function of a discrete random vector (Karr,
1993, p. 116)
Let:
• X1 , . . . , Xd be a discrete r.v., with values in the countable sets C1 , . . . , Cd ,
respectively;
• g : IRd → IR be a Borel measurable function either non negative or integrable (i.e.
g(X1 , . . . , Xd ) ∈ L1 ).
Then
E[g(X1 , . . . , Xd )] =
X
...
x1 ∈C
X
g(x1 , . . . , xd ) × P ({X1 = x1 , . . . , Xd = xd }). (4.56)
xd ∈C
•
Theorem 4.83 — Expectation of a function of an absolutely continuous
random vector (Karr, 1993, p. 116)
Let:
• (X1 , . . . , Xd ) be an absolutely continuous random vector with joint p.d.f.
fX1 ,...,Xd (x1 , . . . , xd );
• g : IRd → IR be a Borel measurable function either non negative or integrable.
Then
E[g(X1 , . . . , Xd )]
Z +∞
Z
=
...
−∞
+∞
g(x1 , . . . , xd ) × fX1 ,...,Xd (x1 , . . . , xd ) dx1 . . . dxd . (4.57)
−∞
•
Exercise 4.84 — Expectation of a function of an absolutely continuous random
vector
Prove Theorem 4.83 (Karr, 1993, pp. 116–117).
•
169
4.3.6
Functions of independent r.v.
When all the components of the random vector (X1 , . . . , Xd ) are independent, the formula
of E[g(X1 , . . . , Xd )] can be simplified.
The next results refer to two independent random variables (d = 2). The generalization
for d > 2 is straightforward.
Theorem 4.85 — Expectation of a function of two independent r.v. (Karr, 1993,
p. 117)
Let:
• X and Y be two independent r.v.
• g : IR2 → IR+ be a Borel measurable non negative function.
Then
g(x, y) dFX (x) dFY (y)
E[g(X, Y )] =
IR
IR
Z Z
=
g(x, y) dFY (y) dFX (x).
Z Z
IR
(4.58)
IR
•
Moreover, the expectation of the product of functions of independent r.v. is the product
of their expectations. Also note that the product of two integrable r.v. need not be
integrable.
Corollary 4.86 — Expectation of a function of two independent r.v. (Karr,
1993, p. 118)
Let:
• X and Y be two independent r.v.
• g1 , g2 : IR → IR be two Borel measurable functions either non negative or integrable.
Then g1 (X) × g2 (Y ) is integrable and
E[g1 (X) × g2 (Y )] = E[g1 (X)] × E[g2 (Y )].
(4.59)
•
Exercise 4.87 — Expectation of a function of two independent r.v.
Prove Theorem 4.85 (Karr, 1993, p. 117) and Corollary 4.86 (Karr, 1993, p. 118).
170
•
4.3.7
Sum of independent r.v.
We are certainly not going to state that E(X + Y ) = E(X) + E(Y ) when X and Y are
simple or non negative or integrable independent r.v.15
Instead, we are going to write the d.f. of the sum of two independent r.v. in terms
of integrals with respect to the d.f. and define a convolution of d.f.16
Theorem 4.88 — D.f. of a sum of two independent r.v. (Karr, 1993, p. 118)
Let X and Y be two independent r.v. Then
Z
Z
FX (t − y) dFY (y) =
FY (t − x) dFX (x).
(4.60)
FX+Y (t) =
IR
IR
•
Exercise 4.89 — D.f. of a sum of two independent r.v.
Prove Theorem 4.88 (Karr, 1993, p. 118).
Corollary 4.90 — D.f. of a sum of two independent discrete r.v.
Let X and Y be two independent discrete r.v. Then
X
X
FY (t − x) × P ({X = x}).
FX (t − y) × P ({Y = y}) =
FX+Y (t) =
•
(4.61)
x
y
•
Remark 4.91 — D.f. of a sum of two independent discrete r.v.
The previous formula is not preferable to the one we derived for the p.f. of X + Y in
Chapter 2 because it depends in fact on two sums...
•
Corollary 4.92 — D.f. of a sum of two independent absolutely continuous r.v.
Let X and Y be two independent absolutely continuous r.v. Then
Z +∞
Z +∞
FX+Y (t) =
FX (t − y) × fY (y) dy =
FY (t − x) × fX (x) dx.
(4.62)
−∞
−∞
•
Let us revisit an exercise from Chapter 2 to illustrate the use of the Corollary 4.92.
15
This result follows from the linearity of expectation.
Recall that in Chapter 2 we derived expressions for the p.f. and the p.d.f. of the sum of two
independent r.v.
16
171
Exercise 4.93 — D.f. of the sum of two independent absolutely continuous r.v.
Let X and Y be the durations of two independent system components set in what is called
a stand by connection.17 In this case the system duration is given by X + Y .
(a) Derive the d.f. of X + Y , assuming that X ∼ Exponencial(α) and Y
Exponencial(β), where α, β > 0 and α 6= β, and using Corollary 4.92.
αβ (e−βz −e−αz )
,
α−β
∼
z > 0.
•
Definition 4.94 — Convolution of d.f. (Karr, 1993, p. 119)
Let X and Y be independent r.v. Then
Z
Z
(FX ? FY )(t) = FX+Y (t) =
FX (t − y) dFY (y) =
FY (t − x) dFX (x)
(4.63)
(b) Prove that the associated p.d.f. equals fX+Y (z) =
IR
IR
is said to be the convolution of the d.f. FX and FY .
17
•
At time 0, only the component with duration X is on. The component with duration Y replaces the
other one as soon as it fails.
172
4.4
Lp spaces
Motivation 4.95 — Lp spaces (Karr, 1993, p. 119)
While describing a r.v. in a partial way, we tend to deal with E(X p ), p ∈ [1, +∞), or a
function of several such expected values. Needless to say that we have to guarantee that
E(|X|p ) is finite.
•
Definition 4.96 — Lp spaces (Karr, 1993, p. 119)
The space Lp , for a fixed p ∈ [1, +∞), consists of all r.v. X whose pth absolute power is
integrable, that is,
E(|X|p ) < +∞.
(4.64)
•
Exercise 4.97 — Exponential distributions and Lp spaces
Let X ∼ Exponential(λ), λ > 0.
Prove that X ∈ Lp , for any p ∈ [1, +∞).
Exercise 4.98 — Pareto distributions and Lp spaces
Let X ∼ Pareto(b, α) i.e.
(
α bα
, x≥b
xα+1
fX (x) =
0,
otherwise
•
(4.65)
where b > 0 is the minimum possible value of X and α > 0 is called the Pareto index.
For which values of p ∈ [1, +∞) we have X ∈ Lp ?
•
173
4.5
Key inequalities
What immediately follows is a table with an overview of a few extremely useful inequalities
involving expectations.
Some of these inequalities are essential to prove certain types of convergence of
sequences of r.v. in Lp and uniform integrability (Resnick, 1999, p. 189)18 and provide
answers to a few questions we compiled after the table.
Finally, we state and treat each inequality separately.
Proposition 4.99 — A
http://en.wikipedia.org)
(Moment) inequality
few
(moment)
inequalities (Karr,
Conditions
IR0+
1993,
p. 123;
Statement of the inequality
IR0+
Young
h:
→
continuous, strictly increasing,
R
h(0) = 0, h(+∞) = +∞, H(x) = 0x h(y) dy
Rx
k pointwise inverse of h, K(x) = 0 k(y) dy
a, b ∈ IR+
a × b ≤ H(a) + K(b)
Hölder
X ∈ Lp , Y ∈ Lq ,
where p, q ∈ [1, +∞) :
E(|X × Y |) ≤ E p (|X|p ) × E q (|Y |q )
(X × Y ∈ L1 )
p
E(|X × Y |) ≤ E(X 2 ) × E(Y 2 )
1
(X × Y ∈ L )
1
1
p
+
1
q
=1
1
Cauchy-Schwarz
X, Y ∈ L2
Liapunov
X ∈ Ls , 1 ≤ r ≤ s
E r (|X|r ) ≤ E s (|X|s )
(Ls ⊆ Lr )
Minkowski
X, Y ∈ Lp , p ∈ [1, +∞)
E p (|X + Y |p ) ≤ E p (|X|p ) + E p (|Y |p )
(X + Y ∈ Lp )
Jensen
g convex; X, g(X) ∈ L1
g[E(X)] ≤ E[g(X)]
g concave; X, g(X) ∈
Chebyshev
1
1
1
L1
1
1
g[E(X)] ≥ E[g(X)]
X ≥ 0,
P ({X ≥ a}) ≤
E[g(X)]
g(a)
E(etX )
eta
g non negative and increasing, a > 0
(Chernoff)
X ≥ 0, a, t > 0
P ({X ≥ a}) ≤
(Markov)
X ∈ L1 , a > 0
P ({|X| ≥ a}) ≤
X ∈ Lp , a > 0
P ({|X| ≥ a}) ≤
X ∈ L2 , a > 0
P ({|X − E(X)| ≥ a}) ≤ a2
“n
o”
p
P
|X − E(X)| ≥ a V (X)
≤
(Chebyshev-Bienaymé)
V (X)
X ∈ L2 , a > 0
(Cantelli)
X ∈ L2 , a > 0
(one-sided Chebyshev)
X ∈ L2 , a > 0
E[|X|]
;
a
E[|X|p ]
ap
1
a2
2V (X)
P ({|X − E(X)| ≥ a}) ≤ a2 +V (X)
“n
o”
p
1
P
X − E(X) ≥ a V (X)
≤ 1+a
2
•
18
For a definition of uniform integrability see http://en.wikipedia.org/wiki/Uniform integrability.
174
Motivation 4.100 — A few (moment) inequalities
• Young — How can we relate the areas under (resp. above) an increasing function h
in the interval [0, a] (resp. in the interval of [0, h−1 (b)]) with the area of the rectangle
with vertices (0, 0), (0, b), (a, 0) and (a, b), where b ∈ (0, maxx∈[0,a] h(x)]?
• Hölder/Cauchy-Schwarz — What are the sufficient conditions on r.v. X and Y
to be dealing with an integrable product XY ?
• Liapunov — What happens to the spaces Lp when p increases in [1, +∞)? Is it a
decreasing (increasing) sequence of sets?
1
What happens to the norm of a r.v. in Lp , ||X||p = E p (|X|p )? Is it an increasing
(decreasing) function of p ∈ [1, +∞)?
• Minkowski — What are the sufficient conditions on r.v. X and Y to be dealing
with a sum X + Y ∈ Lp ? Is Lp a vector space?
• Jensen — Under what conditions we can relate g[E(X)] and E[g(X)]?
• Chebyshev — When can we provide non trivial upper bounds for the tail
probability P ({X ≥ a})
•
4.5.1
Young’s inequality
The first inequality (not a moment inequality) is named after William Henry Young
(1863–1942), an English mathematician, and can be used to prove Hölder inequality.
Lemma 4.101 — Young’s inequality (Karr, 1993, p. 119)
Let:
• h : IR0+ → IR0+ be a continuous and strictly increasing function such that h(0) = 0,
h(+∞) = +∞;
• k be the pointwise inverse of h;
Rx
• H(x) = 0 h(y) dy be the area under h in the interval [0, x];
Rx
• K(x) = 0 k(y) dy be the area above h in the interval [0, h−1 (x)] = [0, k(x)];
• a, b ∈ IR+ .
Then
a × b ≤ H(a) + K(b).
(4.66)
•
175
Exercise 4.102 — Young’s inequality (Karr, 1993, p. 119)
Prove Lemma 4.101, by using a graphical argument (Karr, 1993, p. 119).
•
Remark 4.103 — A special case of Young’s inequality
If we apply Young’s inequality to h(x) = xp−1 , p ∈ [1, +∞) and consider
• a and b non negative real numbers,
• q =1+
1
p−1
∈ [1, +∞) i.e.
1
p
+
1
q
= 1,
then
a×b≤
ap b q
+ .
p
q
(4.67)
For the proof of this result see http://en.wikipedia.org/wiki/Young’s inequality, which
states (4.67) as Young’s inequality. See also Karr (1993, p. 120) for a reference to (4.67)
as a consequence of Young’s inequality as stated in (4.66).
•
4.5.2
Hölder’s moment inequality
In mathematical analysis Hölder’s inequality, named after the German mathematician
Otto Hölder (1859–1937), is a fundamental inequality between integrals, an indispensable
tool for the study of Lp spaces and essential to prove Liapunov’s and Minkowski’s
inequalities.
Interestingly enough, Hölder’s inequality was first found by the British mathematician
L.J. Rogers (1862–1933) in 1888, and discovered independently by Hölder in 1889.
Theorem 4.104 — Hölder’s moment inequality (Karr, 1993, p. 120)
Let
• X ∈ Lp , Y ∈ Lq , where p, q ∈ [1, +∞) :
1
p
+
1
q
= 1.
Then
X × Y ∈ L1
(4.68)
1
p
1
q
E(|XY |) ≤ E (|X|p ) × E (|Y |q ).
(4.69)
•
176
Remarks 4.105 — Hölder’s (moment) inequality
• The numbers p and q above are said to be Hölder conjugates of each other.
• For the detailed statement of Hölder inequality in measure spaces, check
Two notable special cases
http://en.wikipedia.org/wiki/Hölder’s inequality.
follow...
• In case we are dealing with S, a measurable subset of IR with the Lebesgue measure,
and f and g are measurable real-valued functions on S then Hölder’s inequality reads
as follows:
Z
Z
|f (x) × g(x)| dx ≤
S
p1
p
|f (x)| dx
1q
|g(x)| dx .
Z
q
×
S
(4.70)
S
• When we are dealing with n−dimensional Euclidean space and the counting
measure, we have
n
X
k=1
|xk × yk | ≤
n
X
! p1
|xk |p
n
X
×
k=1
! 1q
|yk |q
,
(4.71)
k=1
for all (x1 , . . . , xn ), (y1 , . . . , yn ) ∈ IRn .
• For a generalization of Hölder’s inequality involving n (instead of 2) Hölder
conjugates, see http://en.wikipedia.org/wiki/Hölder’s inequality.
•
Exercise 4.106 — Hölder’s moment inequality
Prove Theorem 4.104, by using the special case of Young’s inequality (4.67), considering
and b = 1 |Y | , taking expectations to (4.67), and applying the result
a = 1 |X|
E p (|X|p )
1
p
+
1
q
E q (|Y |q )
•
= 1 (Karr, 1993, p. 120).
177
4.5.3
Cauchy-Schwarz’s moment inequality
A special case of Hölder’s moment inequality — p = q = 2 — is nothing but the CauchySchwarz’s moment inequality.
In mathematics, the Cauchy-Schwarz inequality19 is a useful inequality encountered
in many different settings, such as linear algebra applied to vectors, in analysis applied to
infinite series and integration of products, and in probability theory, applied to variances
and covariances.
The inequality for sums was published by Augustin Cauchy in 1821, while the
corresponding inequality for integrals was first stated by Viktor Yakovlevich Bunyakovsky
in 1859 and rediscovered by Hermann Amandus Schwarz in 1888 (often misspelled
“Schwartz”).
Corollary 4.107 — Cauchy-Schwarz’s moment inequality (Karr, 1993, p. 120)
Let X, Y ∈ L2 . Then
X × Y ∈ L1
(4.72)
p
E(|X × Y |) ≤ E(|X|2 ) × E(|Y |2 ).
(4.73)
•
Remarks 4.108 — Cauchy-Schwarz’s moment inequality
(http://en.wikipedia.org/wiki/Cauchy-Schwarz inequality)
• In the Euclidean space IRn with the standard inner product, the Cauchy-Schwarz’s
inequality is
n
X
i=1
!2
xi × y i
≤
n
X
!
x2i
i=1
×
n
X
!
yi2 .
(4.74)
i=1
• The triangle inequality for the inner product is often shown as a consequence of the
Cauchy-Schwarz inequality, as follows: given vectors x and y, we have
kx + yk2 = hx + y, x + yi
≤ (kxk + kyk)2 .
(4.75)
•
19
Also known as the Bunyakovsky inequality, the Schwarz inequality, or the Cauchy-BunyakovskySchwarz inequality (http://en.wikipedia.org/wiki/Cauchy-Schwarz inequality).
178
Exercise 4.109 — Confronting the squared covariance and the product of the
variance of two r.v.
Prove that
|cov(X, Y )|2 ≤ V (X) × V (Y ),
(4.76)
where X, Y ∈ L2 (http://en.wikipedia.org/wiki/Cauchy-Schwarz’s inequality).
4.5.4
•
Lyapunov’s moment inequality
1
The spaces Lp decrease as p increases in [1, +∞). Moreover, E(|X|p ) p is an increasing
function of p ∈ [1, +∞).
The following moment inequality is a special case of Hölder’s inequality and is due to
Aleksandr Mikhailovich Lyapunov (1857–1918), a Russian mathematician, mechanician
and physicist.20
Corollary 4.110 — Lyapunov’s moment inequality (Karr, 1993, p. 120)
Let X ∈ Ls , where 1 ≤ r ≤ s. Then
Ls ⊆ Lr
(4.77)
1
s
1
r
E (|X|r ) ≤ E (|X|s ).
(4.78)
•
Remarks 4.111 — Lyapunov’s moment inequality
• Taking s = 2 and r = 1 we can conclude that
E 2 (|X|) ≤ E(X 2 ).
(4.79)
This result is not correctly stated in Karr (1993, p. 121) and can be also deduced
from the Cauchy-Schwarz’s inequality, as well as from Jensen’s inequality, stated
below.
• Rohatgi (1976, p. 103) states Liapunov’s inequality in a slightly different way. It
can be put as follows: for X ∈ Lk , k ∈ [1, +∞),
1
1
E k (|X|k ) ≤ E k+1 (|X|k+1 ).
(4.80)
20
Lyapunov is known for his development of the stability theory of a dynamical system,
as well as for his many contributions to mathematical physics and probability theory
(http://en.wikipedia.org/wiki/Aleksandr Lyapunov).
179
d
• The equality in (4.78) holds iff X is a degenerate r.v., i.e. X = c, where c is a real
constant (Rohatgi, 1976, p. 103).
•
Exercise 4.112 — Lyapunov’s moment inequality
Prove Corollary 4.110, by applying Hölder’s inequality to X 0 = X r , where X ∈ Lr , and
d
•
to Y = 1, and considering p = rs (Karr, 1993, pp. 120–121).21
4.5.5
Minkowski’s moment inequality
The Minkowski’s moment inequality establishes that the Lp spaces are vector spaces.
Theorem 4.113 — Minkowski’s moment inequality (Karr, 1993, p. 121)
Let X, Y ∈ Lp , p ∈ [1, +∞). Then
X + Y ∈ Lp
(4.81)
1
p
1
p
1
p
E (|X + Y |p ) ≤ E (|X|p ) + E (|Y |p ).
(4.82)
•
Remarks 4.114 — Minkowski’s moment inequality
(http://en.wikipedia.org/wiki/Minkowski inequality)
• The Minkowski inequality is the triangle inequality22 in Lp .
• Like Hölder’s inequality, the Minkowski’s inequality can be specialized to (sequences
and) vectors by using the counting measure:
n
X
k=1
! p1
|xk + yk |p
≤
n
X
! p1
|xk |p
k=1
+
n
X
! p1
|yk |p
,
(4.83)
k=1
for all (x1 , . . . , xn ), (y1 , . . . , yn ) ∈ IRn .
•
Exercise 4.115 — Minkowski’s moment inequality
Prove Theorem 4.113, by applying the triangle inequality followed by Hölder’s inequality
and the fact that q(p − 1) = p and 1 − 1q = p1 (Karr, 1993, p. 121).
•
21
Rohatgi (1976, p. 103) provides an alternative proof.
The real line is a normed vector space with the absolute value as the norm, and so the triangle
inequality states that |x + y| ≤ |x| + |y|, for any real numbers x and y. The triangle inequality is useful
in mathematical analysis for determining the best upper estimate on the size of the sum of two numbers,
in terms of the sizes of the individual numbers (http://en.wikipedia.org/wiki/Triangle inequality).
22
180
4.5.6
Jensen’s moment inequality
Jensen’s inequality, named after the Danish mathematician and engineer Johan Jensen
(1859–1925), relates the value of a convex function of an integral to the integral of the
convex function. It was proved by Jensen in 1906.
Given its generality, the inequality appears in many forms depending on the context.
In its simplest form the inequality states, that
• the convex transformation of a mean is less than or equal to the mean after convex
transformation.
It is a simple corollary that the opposite is true of concave transformations
(http://en.wikipedia.org/wiki/Jensen’s inequality).
Theorem 4.116 — Jensen’s moment inequality (Karr, 1993, p. 121)
Let g convex and assume that X, g(X) ∈ L1 . Then
g[E(X)] ≤ E[g(X)]
(4.84)
•
Corollary 4.117 — Jensen’s moment inequality for concave functions
Let g concave and assume that X, g(X) ∈ L1 . Then
g[E(X)] ≥ E[g(X)]
(4.85)
•
Remarks 4.118 — Jensen’s (moment) inequality
(http://en.wikipedia.org/wiki/Jensen’s inequality)
• A proof of Jensen’s inequality can be provided in several ways. However, it is worth
analyzing an intuitive graphical argument based on the probabilistic case where X
is a real r.v.
Assuming a hypothetical distribution of X values, one can immediately identify the
position of E(X) and its image g[E(X)] = ϕ[E(X)] in the graph.
Noticing that for convex mappings Y = g(X) = ϕ(X) the corresponding
distribution of Y values is increasingly “stretched out” for increasing values of X,
the expectation of Y = g(X) will always shift upwards with respect to the position
of g[E(X)] = ϕ[E(X)], and this “proves” the inequality.
181
• For a real convex function g, numbers x1 , x2 , . . . , xn in its domain, and positive
weights ai , i = 1, . . . , n, Jensen’s inequality can be stated as:
Pn
Pn
a × g(xi )
ai × xi
i=1
Pn
Pni
≤ i=1
.
(4.86)
g
i=1 ai
i=1 ai
The inequality is reversed if g is concave.
• As a particular case, if the weights ai = 1, i = 1, . . . , n, then
!
n
n
X
1X
1
g
xi ≤
g(xi ) ⇔ g(x̄) ≤ g(x).
n i=1
n i=1
(4.87)
• For instance, considering g(x) = log(x), which is a concave function, we can establish
the arithmetic mean-geometric mean inequality:23 for any list of n non negative real
numbers x1 , x2 , . . . , xn ,
√
x1 + x2 + . . . + xn
x̄ =
(4.88)
≥ n x1 × x2 × . . . × xn = mg.
n
Moreover, equality in (4.88) holds iff x1 = x2 = . . . = xn .
•
Exercise 4.119 — Jensen’s moment inequality (for concave functions)
Prove Theorem 4.116 (Karr, 1993, pp. 121-122), Corollary 4.117 and Equation (4.88). •
Exercise 4.120 — Jensen’s inequality and the distance between the mean and
the median
Prove that for any r.v. having an expected value and a median, the mean and the median
can never differ from each other by more than one standard deviation:
p
|E(X) − med(X)| ≤ V (X),
(4.89)
by using Jensen’s inequality twice — applied to the absolute value function and to the
square root function24 (http://en.wikipedia.org/wiki/Chebyshev’s inequality).
•
23
24
See http://en.wikipedia.org/wiki/AM-GM inequality.
In this last case we should apply the concave version of Jensen’s inequality.
182
4.5.7
Chebyshev’s inequality
Curiously, Chebyshev’s inequality is named after the Russian mathematician
Pafnuty Lvovich Chebyshev (1821–1894), although it was first formulated by
his friend and French colleague Irénée-Jules Bienaymé (1796–1878), according to
http://en.wikipedia.org/wiki/Chebyshev’s inequality.
In probability theory, the Chebyshev’s inequality,25 in the most usual version — what
Karr (1993, p. 122) calls the Bienaymé-Chebyshev’s inequality —, can be ultimately stated
as follows:
• no more than k12 × 100% of the values of the r.v. X are more than k standard
deviations away from the expected value of X.
Theorem 4.121 — Chebyshev’s inequality (Karr, 1993, p. 122)
Let:
• X be a non negative r.v.;
• g non negative and increasing function on IR+ ;
• a > 0.
Then
P ({X ≥ a}) ≤
E[g(X)]
.
g(a)
(4.90)
•
Exercise 4.122 — Chebyshev’s inequality
Prove Theorem 4.121 (Karr, 1993, p. 122).
•
Remarks 4.123 — Several cases of Chebyshev’s inequality (Karr, 1993, p. 122)
• Chernoff ’s inequality26
X ≥ 0, a, t > 0 ⇒ P ({X ≥ a}) ≤
E(etX )
eta
• Markov’s inequalities
X ∈ L1 , a > 0 ⇒ P ({|X| ≥ a}) ≤
X ∈ Lp , a > 0 ⇒ P ({|X| ≥ a}) ≤
E[|X|]
a
E[|X|p ]
ap
25
Also known as Tchebysheff’s inequality, Chebyshev’s theorem, or the Bienaymé-Chebyshev’s
inequality (http://en.wikipedia.org/wiki/Chebyshev’s inequality).
26
Karr
(1993)
does
not
mention
this
inequality.
For
more
details
see
http://en.wikipedia.org/wiki/Chernoff’s inequality.
183
• Chebyshev-Bienaymé’s inequalities
X ∈ L2 , a > 0 ⇒ P ({|X − E(X)| ≥ a}) ≤ V a(X)
2
n
o
p
X ∈ L2 , a > 0 ⇒ P
|X − E(X)| ≥ a V (X)
≤
1
a2
• Cantelli’s inequality
X ∈ L2 , a > 0 ⇒ P ({|X − E(X)| ≥ a}) ≤
2V (X)
a2 +V (X)
• One-sided Chebyshev’s inequality
n
o
p
X ∈ L2 , a > 0 ⇒ P
X − E(X) ≥ a V (X)
≤
1
1+a2
According to http://en.wikipedia.org/wiki/Chebyshev’s inequality, the one-sided
version of the Chebyshev inequality is also called Cantelli’s inequality, and is due
to the Italian mathematician Francesco Paolo Cantelli (1875–1966).
•
Remark 4.124 — Chebyshev(-Bienaymé)’s inequality
(http://en.wikipedia.org/wiki/Chebyshev’s inequality)
The Chebyshev(-Bienaymé)’s inequality can be useful despite loose bounds because
it applies to random variables of any distribution, and because these bounds can be
calculated knowing no more about the distribution than the mean and variance.
•
Exercise 4.125 — Chebyshev(-Bienaymé)’s inequality
Assume that we have a large body of text, for example articles from a publication and that
we know that the articles are on average 1000 characters long with a standard deviation
of 200 characters.
(a) Prove that from the Chebyshev(-Bienaymé)’s inequality we can then infer that the
chance that a given article is between 600 and 1400 characters would be at least
75%.
(b) The inequality is coarse: a more accurate guess would be possible if the distribution
of the length of the articles is known.
Demonstrate that, for example, a normal distribution would yield a 75% chance of
an article being between 770 and 1230 characters long.
(http://en.wikipedia.org/wiki/Chebyshev’s inequality.)
184
•
Exercise 4.126 — Chebyshev(-Bienaymé)’s inequality
Let X ∼ Uniform(0, 1).
n
o
p
1
(a) Obtain P
|X − 2 | < 2 1/12 .
(b) Obtain a lower bound for P
V (X) =
1
.
12
o
n
p
|X − 21 | < 2 1/12 , by noting that E(X) =
1
2
and
Compare this bound with the value you obtained in (a).
•
(Rohatgi, 1976, p. 101.)
Exercise 4.127 — Meeting the Chebyshev(-Bienaymé)’s bounds exactly
Typically, the Chebyshev(-Bienaymé)’s inequality will provide rather loose bounds.
(a) Prove that these bounds cannot be improved upon for the r.v. X with p.f.

P ({X = −1}) = 2k12 ,



 P ({X = 0}) = 1 − 1 ,
k2
P ({X = x}) =
1

P
({X
=
1})
=
,

2k2


0,
x = −1
x=0
x=1
otherwise,
p
where k > 1, that is, P |X − E(X)| ≥ k V (X) =
http://en.wikipedia.org/wiki/Chebyshev’s inequality.)27
1
.
k2
(4.91)
(For more details see
(b) Prove that equality holds exactly for any r.v. Y that is a linear transformation of
X.28
•
Remark 4.128 — Chebyshev(-Bienaymé)’s inequality and the weak law of
large numbers (http://en.wikipedia.org/ wiki/Chebyshev’s inequality)
Chebyshev(-Bienaymé)’s inequality is used for proving the following version of the weak
law of large numbers: when dealing with a sequence of i.i.d. r.v., X1 , X2 , . . ., with finite
expected value and variance (µ, σ 2 < +∞),
lim P
X̄n − µ < = 1,
(4.92)
n→+∞
where X̄n =
1
n
Pn
i=1
P
Xi . That is, X̄n → µ as n → +∞.
•
27
This is the answer to Exercise 4.36 from Karr (1993, p. 133).
Inequality holds for any r.v. that is not a
(http://en.wikipedia.org/wiki/Chebyshev’s inequality).
28
185
linear
transformation
of
X
Exercise 4.129 — Chebyshev(-Bienaymé)’s inequality and the weak law of
large numbers
Use Chebyshev(-Bienaymé)’s inequality to prove the weak law of large numbers stated in
Remark 4.128.
•
Exercise 4.130 — Cantelli’s inequality (Karr, 1993, p. 132, Exercise 4.30)
(X)
give a better bound than Chebyshev(When does P ({|X − E(X)| ≥ a}) ≤ a22V
+V (X)
Bienaymé)’s inequality?
•
Exercise 4.131 — One-sided Chebyshev’s inequality and the distance between
the mean and the median
Use the one-sided Chebyshev’s inequality to prove that for any r.v. having an expected
value and a median, the mean and the median can never differ from each other by more
than one standard deviation, i.e.
p
(4.93)
|E(X) − med(X)| ≤ V (X)
(http://en.wikipedia.org/wiki/Chebyshev’s inequality).
186
•
4.6
Moments
Motivation 4.132 — Moments of r.v.
The nature of a r.v. can be partial described in terms of a number of features — such
as the expected value, the variance, the skewness, kurtosis, etc. — that can written in
terms of expectation of powers of X, the moments of a r.v.
•
4.6.1
Moments of r.v.
Definition 4.133 — kth. moment and kth. central moment of a r.v. (Karr, 1993,
p. 123)
Let
• X be a r.v. such that X ∈ Lk , for some k ∈ IN .
Then:
• the kth. moment of X is given by the Riemann-Stieltjes integral
Z ∞
k
xk dFX (x);
E(X ) =
(4.94)
−∞
• similarly, the kth. central moment of X equals
Z +∞
k
E [X − E(X)] =
[x − E(X)]k dFX (x).
−∞
(4.95)
•
Remarks 4.134 — kth. moment and kth. central moment of a r.v. (Karr, 1993,
p. 123; http://en.wikipedia.org/wiki/Moment (mathematics))
• The kth. central moment exists under the assumption that X ∈ Lk because Lk ⊆ L1 ,
for any k ∈ IN (a consequence of Lyapunov’s inequality).
• If the kth. (central) moment exists29 so does the (k − 1)th. (central) moment, and
all lower-order moments. This is another consequence of Lyapunov’s inequality.
• If X ∈ L1 the first moment is the expectation of X; the first central moment is thus
null. In higher orders, the central moments are more interesting than the moments
about zero.
•
29
Or the kth. moment about any point exists. Note that the kth. central moment is nothing but the
kth. moment about E(X).
187
Proposition 4.135 — Computing the kth. moment of a non negative r.v.
If X is a non negative r.v. and X ∈ Lk , for k ∈ IN , we can write the kth. moment of X
in terms of the following Riemann integral:
Z ∞
k
E(X ) =
k xk−1 × [1 − FX (x)] dx.
(4.96)
0
•
Exercise 4.136 — Computing the kth. moment of a non negative r.v.
Prove Proposition 4.135.
•
Exercise 4.137 — Computing the kth. moment of a non negative r.v.
, for any
Let X ∼ Exponential(λ). Use Proposition 4.135 to prove that E(X k ) = Γ(k+1)
λk
k ∈ IN .
•
Exercise 4.138 — The median of a r.v. and the minimization of the expected
absolute deviation (Karr, 1993, p. 130, Exercise 4.1)
The median of the r.v. X, med(X), is such that P ({X ≤ med(X)}) ≥ 21 and P ({X ≥
med(X)}) ≥ 12 .
Prove that if X ∈ L1 then
E(|X − med(X)|) ≤ E(|X − a|),
(4.97)
for all a ∈ IR.
•
Exercise 4.139 — Minimizing the mean squared error (Karr, 1993, p. 131,
Exercise 4.12)
Let {A1 , . . . , An } be a finite partition of Ω. Suppose that we know which of A1 , . . . , An
has occurred, and wish to predict whether some other event B has occurred. Since we
know the values of the indicator functions 1A1 , . . . , 1An , it make sense to use a predictor
P
that is a function of them, namely linear predictors of the form Y = ni=1 ai × 1Ai , whose
accuracy is assessed via the mean squared error:
MSE(Y ) = E[(1B − Y )2 ].
(4.98)
Prove that the values ai = P (B|Ai ), i = 1, . . . , n minimize MSE(Y ).
188
•
Exercise 4.140 — Expectation of a r.v. with respect to a conditional
probability function (Karr, 1993, p. 132, Exercise 4.20)
Let A be an event such that P (A) > 0.
Show that if X is positive or integrable then E(X|A), the expectation of X with
respect to the conditional probability function PA (B) = P (B|A), is given by
def
E(X|A) =
E(X; A)
,
P (A)
(4.99)
where E(X; A) = E(X × 1A ) represents the expectation of X over the event A.
4.6.2
•
Variance and standard deviation
Definition 4.141 — Variance and standard deviation of a r.v. (Karr, 1993, p.
124)
Let X ∈ L2 . Then:
• the 2nd. central moment is the variance of X,
V (X) = E {[ X − E(X)]2 };
(4.100)
• the positive square root of the variance is the standard deviation of X,
p
SD(X) = + V (X).
(4.101)
•
Remark 4.142 — Computing the variance of a r.v. (Karr, 1993, p. 124)
The variance of a r.v. X ∈ L2 can also be expressed as
V (X) = E(X 2 ) − E 2 (X),
(4.102)
which is more convenient than (4.100) for computational purposes.
•
Exercise 4.143 — The meaning of a null variance (Karr, 1993, p. 131, Exercise
4.19)
a.s.
Prove that if V (X) = 0 then X = E(X).
•
Exercise 4.144 — Comparing the variance of X and min{X, c} (Karr, 1993, p.
133, Exercise 4.32)
Let X be a r.v. such that E(X 2 ) < +∞ and c a real constant. Prove that
V (min{X, c}) ≤ V (X).
(4.103)
•
189
Proposition 4.145 — Variance of the sum (or difference) of two independent
r.v. (Karr, 1993, p. 124)
If X, Y ∈ L2 and are two independent then
V (X + Y ) = V (X − Y ) = V (X) + V (Y ).
(4.104)
•
Exercise 4.146 — Expected values and variances of some important r.v. (Karr,
1993, pp. 125 and 130, Exercise 4.1)
Verify the entries of the following table.
Distribution
Parameters
Expected value
Discrete Uniform({x1 , x2 , . . . , xn })
{x1 , x2 , . . . , xn }
1
n
Bernoulli(p)
p ∈ [0, 1]
p
p (1 − p)
Binomial(n, p)
n ∈ IN ; p ∈ [0, 1]
np
n p (1 − p)
Hipergeometric(N, M, n)
N ∈ IN
M ∈ IN, M ≤ N
n ∈ IN, n ≤ N
nM
N
nM
1−
N
Geometric(p)
p ∈ [0, 1]
1
p
1−p
p2
Poisson(λ)
λ ∈ IR+
λ
λ
NegativeBinomial(r, p)
r ∈ IN ; p ∈ [0, 1]
r
p
r(1−p)
p2
Uniform(a, b)
a, b ∈ IR, a < b
a+b
2
(b−a)2
12
Normal(µ, σ 2 )
µ ∈ IR; σ 2 ∈ IR+
µ
σ2
Lognormal(µ, σ 2 )
µ ∈ IR; σ 2 ∈ IR+
eµ+ 2 σ
Exponential(λ)
λ ∈ IR+
1
λ
1
λ2
Gamma(α, β)
α, β ∈ IR+
α
β
α
β2
Beta(α, β)
α, β ∈ IR+
α
α+β
αβ
(α+β)2 (α+β+1)
Weibull(α, β)
α, β ∈ IR+
α Γ 1 + β1
h i
α2 Γ 1 + β2 − Γ2 1 + β1
Pn
i=1
1
xi
2
Variance
1
n
Pn
i=1
x2i −
M
N
1
n
Pn
i=1
xi
2
(eσ − 1)e2µ+σ
2
•
190
2
Definition 4.147 — Normalized (central) moments of a r.v.
(http://en.wikipedia.org/wiki/Moment (mathematics))
Let X be a r.v. such that X ∈ Lk , for some k ∈ IN . Then:
• the normalized kth. moment of the X is the kth. moment divided by [SD(X)]k ,
E(X k )
;
[SD(X)]k
(4.105)
• the normalized kth. central moment of X is given by
E{[X − E(X)]k }
;
[SD(X)]k
(4.106)
These normalized central moments are dimensionless quantities, which represent the
distribution independently of any linear change of scale.
•
4.6.3
Skewness and kurtosis
Motivation 4.148 — Skewness of a r.v.
(http://en.wikipedia.org/wiki/Moment (mathematics))
Any r.v. X ∈ L3 with a symmetric p.(d.)f. will have a null 3rd. central moment. Thus,
the 3rd. central moment is a measure of the lopsidedness of the distribution.
•
Definition 4.149 — Skewness of a r.v.
(http://en.wikipedia.org/wiki/Moment (mathematics))
Let X ∈ L3 be a r.v. Then the normalized 3rd. central moment is called the skewness —
or skewness coefficient (SC) —,
SC(X) =
E{[X − E(X)]3 }
.
[SD(X)]3
(4.107)
•
Remark 4.150 — Skewness of a r.v.
(http://en.wikipedia.org/wiki/Moment (mathematics))
• A r.v. X that is skewed to the left (the tail of the p.(d.)f. is heavier on the right)
will have a negative skewness.
• A r.v. that is skewed to the right (the tail of the p.(d.)f. is heavier on the left), will
have a positive skewness.
•
191
Exercise 4.151 — Skewness of a r.v.
Prove that the skewness of:
(a) X ∼ Exponential(λ) equals SC(X) = 2
(http://en.wikipedia.org/wiki/Exponential distribution);
q
α−2
(b) X ∼ Pareto(b, α) is given by SC(X) = 2(1+α)
, for α > 3
α−3
α
(http://en.wikipedia.org/wiki/Pareto distribution).
•
Motivation 4.152 — Kurtosis of a r.v.
(http://en.wikipedia.org/wiki/Moment (mathematics))
The normalized 4th. central moment of any normal distribution is 3. Unsurprisingly, the
normalized 4th. central moment is a measure of whether the distribution is tall and skinny
or short and squat, compared to the normal distribution of the same variance.
•
Definition 4.153 — Kurtosis of a r.v.
(http://en.wikipedia.org/wiki/Moment (mathematics))
Let X ∈ L4 be a r.v. Then the kurtosis — or kurtosis coefficient (KC) — is defined to be
the normalized 4th. central moment minus 3,30
KC(X) =
E{[X − E(X)]4 }
− 3.
[SD(X)]4
(4.108)
•
Remarks 4.154 — Kurtosis of a r.v.
(http://en.wikipedia.org/wiki/Moment (mathematics))
• If the p.(d.)f. of the r.v. X has a peak at the expected value and long tails, the 4th.
moment will be high and the kurtosis positive. Bounded distributions tend to have
low kurtosis.
• KC(X) must be greater than or equal to [SC(X)]2 − 2; equality only holds for
Bernoulli distributions (prove!).
• For unbounded skew distributions not too far from normal, KC(X) tends to be
somewhere between [SC(X)]2 and 2 × [SC(X)]2 .
•
30
Some authors do not subtract three.
192
Exercise 4.155 — Kurtosis of a r.v.
Prove that the kurtosis of:
(a) X ∼ Exponential(λ) equals KC(X) = 6
(http://en.wikipedia.org/wiki/Exponential distribution);
(b) X ∼ Pareto(b, α) is given by
6(α3 +α2 −6α−2)
α(α−3)(α−4)
for α > 4
(http://en.wikipedia.org/wiki/Pareto distribution).
4.6.4
•
Covariance
Motivation 4.156 — Covariance (and correlation) between two r.v.
It is crucial to obtain measures of how much two variables change together, namely
absolute and relative measures of (linear) association between pairs of r.v.
•
Definition 4.157 — Covariance between two r.v. (Karr, 1993, p. 125)
Let X, Y ∈ L2 be two r.v. Then the covariance between X and Y is equal to
cov(X, Y ) = E{[X − E(X)] × [Y − E(Y )]}
= E(XY ) − E(X)E(Y )
(this last formula is more useful for computational purposes, prove it!)
(4.109)
•
Remark 4.158 — Covariance between two r.v.
(http://en.wikipedia.org/wiki/Covariance)
The units of measurement of the covariance between the r.v. X and Y are those of X
times those of Y .
•
Proposition 4.159 — Properties of the covariance
Let X, Y, Z ∈ L2 , X1 , . . . , Xn ∈ L2 , Y1 , . . . , Yn ∈ L2 , and a, b ∈ IR. Then:
1. X⊥
⊥Y ⇒ cov(X, Y ) = 0
2. cov(X, Y ) = 0 6⇒ X⊥
⊥Y
3. cov(X, Y ) 6= 0 ⇒ X ⊥
6 ⊥Y
193
4. cov(X, Y ) = cov(Y, X) (symmetric operator!)
a.s.
5. cov(X, X) = V (X) ≥ 0 and V (X) = 0 ⇒ X = E(X) (positive semi-definite
operator!)
6. cov(aX, bY ) = a b cov(X, Y )
7. cov(X + a, Y + b) = cov(X, Y )
8. cov(aX + bY, Z) = a cov(X, Z) + b cov(Y, Z) (bilinear operator!)
P
P P
Pn
n
9. cov
X
,
Y
= ni=1 nj=1 cov(Xi , Yj )
i
j
i=1
j=1
10. cov
P
n
i=1
Xi ,
P
P P
X
= ni=1 V (Xi ) + 2 × ni=1 nj=i+1 cov(Xi , Xj ).
j
j=1
Pn
Exercise 4.160 — Covariance
Prove properties 6 through 10 from Proposition 4.159.
•
•
Proposition 4.161 — Variance of some linear combinations of r.v.
Let X1 , . . . , Xn ∈ L2 . Then:
V (c1 X1 + c2 X2 ) = c21 V (X1 ) + c22 V (X2 ) + 2c1 c2 cov(X1 , X2 );
V (X1 + X2 ) = V (X1 ) + V (X2 ) + 2cov(X1 , X2 );
V (X1 − X2 ) = V (X1 ) + V (X2 ) − 2cov(X1 , X2 );
!
n
n
n X
n
X
X
X
2
V
ci X i
=
ci V (Xi ) + 2
ci cj cov(Xi , Xj ).
i=1
i=1
(4.110)
(4.111)
(4.112)
(4.113)
i=1 j=i+1
When we deal with uncorrelated r.v. — i.e., if cov(Xi , Xj ) = 0, ∀i 6= j — or with pairwise
independent r.v. — that is, Xi⊥
⊥Xj , ∀i 6= j —, we have:
!
n
n
X
X
ci X i =
c2i V (Xi ).
(4.114)
V
i=1
i=1
And if, besides being uncorrelated or (pairwise) independent r.v., we have ci = 1, for
i = 1, . . . , n, we get:
!
n
n
X
X
Xi =
V (Xi ),
(4.115)
V
i=1
i=1
i.e. the variance of the sum of uncorrelated or (pairwise) independent r.v. is the sum of
the individual variances.
•
194
4.6.5
Correlation
Motivation 4.162 — Correlation between two r.v.
(http://en.wikipedia.org/wiki/Correlation and dependence)
The most familiar measure of dependence between two r.v. is (Pearson’s) correlation.
It is obtained by dividing the covariance between two variables by the product of their
standard deviations.
Correlations are useful because they can indicate a predictive relationship that can
be exploited in practice. For example, an electrical utility may produce less power on a
mild day based on the correlation between electricity demand and weather. Moreover,
correlations can also suggest possible causal, or mechanistic relationships.
•
Definition 4.163 — Correlation between two r.v. (Karr, 1993, p. 125)
Let X, Y ∈ L2 be two r.v. Then the correlation31 between X and Y is given by
cov(X, Y )
.
corr(X, Y ) = p
V (X) V (Y )
(4.116)
•
Remark 4.164 — Correlation between two r.v.
(http://en.wikipedia.org/wiki/Covariance)
Correlation is a dimensionless measure of linear dependence.
•
Definition 4.165 — Uncorrelated r.v. (Karr, 1993, p. 125)
Let X, Y ∈ L2 . Then if
corr(X, Y ) = 0
(4.117)
X and Y are said to be uncorrelated r.v.32
•
Exercise 4.166 — Uncorrelated r.v. (Karr, 1993, p. 131, Exercise 4.14)
Give an example of r.v. X and Y that are uncorrelated but for which there is a function
g such that Y = g(X).
•
Exercise 4.167 — Uncorrelated r.v. (Karr, 1993, p. 131, Exercise 4.18)
d
Prove that if V, W ∈ L2 and (V, W ) = (−V, W ) then V and W are uncorrelated.
31
•
Also know as the Pearson’s correlation coefficient (http://en.wikipedia.org/wiki/Correlation and
dependence).
32
X, Y ∈ L2 are said to be correlated r.v. if corr(X, Y ) 6= 0.
195
Exercise 4.168 — Sufficient conditions to deal with uncorrelated sample mean
and variance (Karr, 1993, p. 132, Exercise 4.28)
i.i.d.
Let Xi ∼ X, i = 1, . . . , n, such that E(X) = E(X 3 ) = 0.
P
Prove that the sample mean X̄ = n1 ni=1 Xi and the sample variance
Pn
1
2
S 2 = n−1
•
i=1 (Xi − X̄) are uncorrelated r.v.
Proposition 4.169 — Properties of the correlation
Let X, Y ∈ L2 , and a, b ∈ IR. Then:
1. X⊥
⊥Y ⇒ corr(X, Y ) = 0
2. corr(X, Y ) = 0 6⇒ X⊥
⊥Y
3. corr(X, Y ) 6= 0 ⇒ X ⊥
6 ⊥Y
4. corr(X, Y ) = corr(Y, X)
5. corr(X, X) = 1
6. corr(aX, bY ) = corr(X, Y )
7. −1 ≤ corr(X, Y ) ≤ 1, for any pair of r.v.33
a.s.
8. corr(X, Y ) = −1 ⇔ Y = aX + b, a < 0
a.s.
9. corr(X, Y ) = 1 ⇔ Y = aX + b, a > 0.
Exercise 4.170 — Properties of the correlation
Prove properties 7 through 9 from Proposition 4.169.
•
•
Exercise 4.171 — Negative linear association between three r.v. (Karr, 1993, p.
131, Exercise 4.17)
Prove that there are no r.v. X, Y and Z such that corr(X, Y ) = corr(Y, Z) =
corr(Z, X) = −1.
•
Remark 4.172 — Interpretation of the sign of a correlation
The correlation sign entre X e Y should be interpreted as follows:
• if corr(X, Y ) is “considerably” larger than zero (resp. smaller than zero) we can
cautiously add that if X increases then Y “tends” to increase (resp. decrease).
•
33
A consequence of Cauchy-Schwarz’s inequality.
196
Remark 4.173 — Interpretation of the size of a correlation
(http://en.wikipedia.org/wiki/Pearson product-moment correlation coefficient)
Several authors have offered guidelines for the interpretation of a correlation coefficient.
Others have observed, however, that all such criteria are in some ways arbitrary and
should not be observed too strictly.
The interpretation of a correlation coefficient depends on the context and purposes.
A correlation of 0.9 may be very low if one is verifying a physical law using high-quality
instruments, but may be regarded as very high in the social sciences where there may be
a greater contribution from complicating factors.
•
Remark 4.174 — Correlation and linearity
(http://en.wikipedia.org/wiki/Correlation and dependence)
Properties 8 and 9 from Proposition 4.169 suggest that
• correlation “quantifies” the linear association between X e Y .
Thus, if the absolute value of corr(X, Y ) is very close to the unit we are tempted to add
that the association between X and Y is “likely” to be linear.
However, the Pearson’s correlation coefficient indicates the strength of a linear
relationship between two variables, but its value generally does not completely characterize
their relationship.
The image on the right shows scatterplots of Anscombe’s quartet, a set of four different
pairs of variables created by Francis Anscombe. The four y variables have the same mean
(7.5), standard deviation (4.12), correlation (0.816) and regression line (y = 3 + 0.5x).
However, as can be seen on the plots, the distribution of the variables is very different.
The first one (top left) seems to be distributed normally, and corresponds to what one
would expect when considering two variables correlated and following the assumption of
normality.
197
The second one (top right) is not distributed normally; while an obvious relationship
between the two variables can be observed, it is not linear, and the Pearson correlation
coefficient is not relevant.
In the third case (bottom left), the linear relationship is perfect, except for one outlier
which exerts enough influence to lower the correlation coefficient from 1 to 0.816.
Finally, the fourth example (bottom right) shows another example when one outlier
is enough to produce a high correlation coefficient, even though the relationship between
the two variables is not linear.
•
Remark 4.175 — Correlation and causality
(http://en.wikipedia.org/wiki/Correlation and dependence)
The conventional dictum that “correlation does not imply causation” means that
correlation cannot be used to infer a causal relationship between the variables.34
This dictum should not be taken to mean that correlations cannot indicate the
potential existence of causal relations. However, the causes underlying the correlation,
if any, may be indirect and unknown, and high correlations also overlap with identity
relations, where no causal process exists. Consequently, establishing a correlation between
two variables is not a sufficient condition to establish a causal relationship (in either
direction).
Several sets of (x, y) points, with the correlation coefficient of x and y for each set.
Note that the correlation reflects the noisiness and direction of a linear relationship (top
row), but not the slope of that relationship (middle), nor many aspects of nonlinear
relationships (bottom). The figure in the center has a slope of 0 but in that case the
correlation coefficient is undefined because the variance of y is zero.
•
34
A correlation between age and height in children is fairly causally transparent, but a correlation
between mood and health in people is less so. Does improved mood lead to improved health; or does
good health lead to good mood; or both? Or does some other factor underlie both? In other words,
a correlation can be taken as evidence for a possible causal relationship, but cannot indicate what the
causal relationship, if any, might be.
198
4.6.6
Moments of random vectors
Moments of random vectors are defined component, pairwise, etc. Instead of an expected
value (resp. variance) we shall deal with a mean vector (resp. covariance matrix).
Definition 4.176 — Mean vector and covariance matrix of a random vector
(Karr, 1993, p. 126)
Let X = (X1 , . . . , Xd ) be a d−dimensional random vector. Then provided that:
• Xi ∈ L1 , i = 1, . . . , d, the mean vector of X is the d−vector of the individual means,
µ = (E(X1 ), . . . , E(Xd ));
• Xi ∈ L2 , i = 1, . . . , d, the covariance matrix of X is a d × d matrix given by
Σ = [cov(Xi , Xj )]i,j=1,...,d .
•
Proposition 4.177 — Properties of the covariance matrix of a random vector
(Karr, 1993, p. 126)
Let X = (X1 , . . . , Xd ) be a d−dimensional random vector with covariance matrix Σ .
Then:
• the diagonal of Σ has entries equal to cov(Xi , Xi ) = V (Xi ), i = 1, . . . , d;
• Σ is a symmetric matrix since cov(Xi , Xj ) = cov(Xj , Xi ), i, j = 1, . . . , d;
P P
• Σ is a positive-definite matrix, that is, di=1 dj=1 xi × cov(Xi , Xj ) × xj > 0, for
every d−vector x = (x1 , . . . , xd ).
•
Exercise 4.178 — Mean vector and covariance matrix of a linear combination
of r.v. (matrix notation)
Let:
• X = (X1 , . . . , Xd ) a d−dimensional random vector;
• µ = (E(X1 ), . . . , E(Xd )) the mean vector of X;
• Σ = [cov(Xi , Xj )]i,j=1,...,n the covariance matrix of X;
• c = (c1 , . . . , cn ) a vector of weights.
P
By noting that ni=1 ci Xi = c> X, verify that:
P
• E ( ni=1 ci Xi ) = c> µ;
P
• V ( ni=1 ci Xi ) = c> Σ c.
199
•
4.6.7
Multivariate normal distributions
Motivation 4.179 — Multivariate normal distribution (Tong, 1990, p. 1)
There are many reasons for the predominance of the multivariate normal distribution:
• it represents a natural extension of the univariate normal distribution and provides
a suitable model for many real-life problems concerning vector-valued data;
• even if the original data cannot be fitted satisfactorily with a multivariate normal
distribution, by the central limit theorem the distribution of the sample mean vector
is asymptotically normal;
• the p.d.f. of a multivariate normal distribution is uniquely determined by the mean
vector and the covariance matrix;
• zero correlation imply independence between two components of the random vector
with multivariate normal distribution;
• the family of multivariate normal distributions is closed under linear transformations
or linear combinations;
• the marginal distribution of any subset of components of a random vector with
multivariate normal distribution;
• the conditional distribution in a multivariate normal distribution is multivariate
normal.
•
Remark 4.180 — Multivariate normal distribution (Tong, 1990, p. 2)
Studies of the bivariate normal distribution seem to begin in the middle of the XIX
century, and moved forward in 1888 with F. Galton’s (1822–1911) work on the applications
of correlations analysis in genetics. In 1896, K. Pearson (1857–1936) gave a definitive
mathematical formulation of the bivariate normal distribution.
The multivariate normal distribution was treated comprehensively for the first time
in 1892 by F.Y. Edgeworth (1845–1926).
•
A random vector has a multivariate normal distribution if it is a linear transformation
of a random vector with i.i.d. components with standard normal distribution (Karr, 1993,
p. 126).
200
Definition 4.181 — Multivariate normal distribution (Karr, 1993, p. 126)
Let:
• µ = (µ1 , . . . , µd ) ∈ IRd ;
• Σ = [σij ]i,j=1,...,d be a symmetric, positive-definite, non-singular d × d matrix;35
Then the random vector X = (X1 , . . . , Xd ) has a multivariate normal distribution with
mean vector µ and covariance matrix Σ if
1
X = Σ 2 Y + µ,
(4.118)
where:
i.i.d.
• Y = (Y1 , . . . , Yd ) with Yi ∼ Normal(0, 1), i = 1, . . . , d;
1 >
1
1
2
• Σ is the unique matrix satisfying Σ 2
× Σ 2 = Σ.
In this case we write X ∼ Normald (µ, Σ ).
•
We can use Definition 4.181 to simulate a multivariate normal distribution as
mentioned below.
Remark 4.182 — Simulating a multivariate normal distribution (Gentle, 1998,
pp. 105–106)
1
i.i.d.
Since Yi ∼ Normal(0, 1), i = 1, . . . , d, implies that X = Σ 2 Y + µ ∼ Normald (µ, Σ )
we can obtain a d−dimensional pseudo-random vector from this multivariate normal
distribution if we:
1. generate d independent pseudo-random numbers, y1 , . . . , yd , from the standard
normal distribution;
1
2. assign x = Σ 2 y + µ, where y = (y1 , . . . , yd ).
Gentle (1998, p. 106) refers other procedures to generate pseudo-random numbers with
multivariate normal distribution.
•
35
A d × d matrix A is called invertible or non-singular or non-degenerate if there exists an
d × d matrix B such that AB = BA = Id , where Id denotes the d × d identity matrix.
(http://en.wikipedia.org/wiki/Invertible matrix) .
201
Proposition 4.183 — Characterization of the multivariate normal distribution
(Karr, 1993, pp. 126–127)
Let X ∼ Normald (µ, Σ ) where µ = (µ1 , . . . , µd ) and Σ = [σi j ]i,j=1,...,d . Then:
E(Xi ) = µi , i = 1, . . . , d;
(4.119)
cov(Xi , Xj ) = σij , i, j = 1, . . . , d;
(4.120)
d
1
1
Σ|− 2 × exp − (x − µ)> Σ −1 (x − µ) ,
fX (x) = (2π)− 2 |Σ
2
for x = (x1 , . . . , xd ) ∈ IRd .
(4.121)
•
Exercise 4.184 — P.d.f. of a bivariate normal distribution
Let (X1 , X2 ) have a (non-singular) bivariate normal distribution with mean vector and
covariance matrix
"
#
"
#
µ1
σ12
ρσ1 σ2
µ=
and Σ =
,
(4.122)
µ2
ρσ1 σ2
σ22
respectively, where |ρ| = |corr(X, Y )| < 1.
(a) Verify that the joint p.d.f. is given by
(
"
2
1
1
x1 − µ 1
p
fX1 ,X2 (x1 , x2 ) =
exp −
2(1 − ρ2 )
σ1
2πσ1 σ2 1 − ρ2
#)
2
x 1 − µ1
x2 − µ 2
x2 − µ 2
−2ρ
+
, (x1 , x2 ) ∈ IR2 . (4.123)
σ1
σ2
σ2
(b) Use Mathematica to plot this joint p.d.f. for µ1 = µ2 = 0 and σ12 = σ22 = 1, and at
least five different values of the correlation coefficient ρ.
•
Exercise 4.185 — Normally distributed r.v. with a non bivariate normal
distribution
We have already mentioned that if two r.v. X1 and X2 both have a standard normal
distribution this does not imply that the random vector (X1 , X2 ) has a joint normal
distribution.36
Prove that X2 = X1 if |X1 | > c and X2 = −X1 if |X1 | < c, where c > 0, illustrates
this fact.
•
36
See http://en.wikipedia.org/wiki/Multivariate normal distribution.
202
In what follows we describe a few distributional properties of bivariate normal
distributions and, more generally, multivariate normal distributions.
Proposition 4.186 — Marginal distributions/moments in the bivariate normal
setting (Tong, 1990, p. 8, Theorem 2.1.1)
Let X = (X1 , X2 ) be distributed according to a bivariate normal distribution with
parameters
"
#
"
#
ρσ1 σ2
µ1
σ12
and Σ =
.
(4.124)
µ=
ρσ1 σ2
σ22
µ2
Then the marginal distribution of Xi , i = 1, 2, is normal. In fact,
Xi ∼ Normal(µi , σi2 ), i = 1, 2.
(4.125)
The following figure37 shows the two marginal distributions of a bivariate normal
distribution:
•
Consider the partitions of X, µ and Σ given below,
"
#
"
#
"
#
µ1
X1
Σ 11 Σ 12
X=
, µ=
and Σ =
,
X2
µ2
Σ 21 Σ 22
(4.126)
where:
• X 1 = (X1 , . . . , Xk ) is made up of the first k < d components of X;
• X 2 = (Xk+1 , . . . , Xd ) is made up of the remaining components of X;
• µ1 = (µ1 , . . . , µk );
• µ2 = (µk+1 , . . . , µd );
37
Taken from http://www.aiaccess.net/English/Glossaries/GlosMod/e gm multinormal distri.htm.
203
• Σ 11 = [σij ]i,j=1,...,k ; Σ 12 = [σij ]1≤i<j≤k ;
• Σ 21 = Σ >
12 ;
• Σ 22 = [σij ]i,j=k+1,...,d .
The following figure (where d = p)38 represents the covariance matrix of X 1 , Σ 11 , which
is just the upper left corner square submatrix of order k of the original covariance matrix:
Theorem 4.187 — Marginal distributions/moments in the multivariate normal
setting (Tong, 1990, p. 30, Theorem 3.3.1)
Let X ∼ Normald (µ, Σ ). Then for every k < d the marginal distributions of X 1 and X 2
are also multivariate normal:
X 1 ∼ Normalk (µ1 , Σ 11 )
(4.127)
X 2 ∼ Normald−k (µ2 , Σ 22 ),
(4.128)
•
respectively.
The family of multivariate normal distributions is closed under linear transformations,
as stated below.
Theorem 4.188 — Distribution/moments of a linear transformation of a
bivariate normal random vector (Tong, 1990, p. 10, Theorem 2.1.2)
Let:
• X ∼ Normal2 (µ, Σ );
• C = [cij ] be a 2 × 2 real matrix;
• b = (b1 , b2 ) be a real vector.
Then
Y = C X + b ∼ Normal2 (C µ + b, C Σ C> ).
(4.129)
•
38
Also taken from http://www.aiaccess.net/English/Glossaries/GlosMod/e gm multinormal distri.htm.
204
Exercise 4.189 — Distribution/moments of a linear transformation of a
bivariate normal random vector
(a) Prove that if in Theorem 4.188 we choose
"
C=
σ1−1 0
0 σ2−1
#
(4.130)
and b = −µC, then Y is a bivariate normal variable with zero means, unit variances
and correlation coefficient ρ (Tong, 1990, p. 10).
(b) Now consider a linear transformation of Y , Y ∗ , by rotating the xy axes by 45 degrees
counterclockwise:
"
#
1
−1
1
Y.
(4.131)
Y∗ = √
2 1 1
Verify that Y ∗ is a bivariate normal variable with zero means, variances 1 − ρ and
1 + ρ and null correlation coefficient (Tong, 1990, p. 10). Comment.
(c) Conclude that if X ∼ Normal2 (µ, Σ ) such that |ρ| < 1 then
"
#"
√1
1−ρ
0
0
√1
1+ρ
√1
2
√1
2
− √12
#"
√1
2
σ1−1 0
0 σ2−1
#"
X1 − µ1
X2 − µ2
#
∼ Normal2 (0, I2 ), (4.132)
where 0 = (0, 0) and I2 is the 2 × 2 identity matrix (Tong, 1990, p. 11).
i.i.d.
(d) Prove that if Zi ∼ Normal(0, 1) then
"
σ1 0
0 σ2
#"
√1
2
− √12
√1
2
√1
2
#" √
1−ρ
0
√
0
1+ρ
#"
Z1
Z2
#
∼ Normal2 (µ, Σ ), (4.133)
i.e. we can obtain a bivariate normal distribution with any mean vector and (nonsingular, semi-definite positive) covariance matrix through a transformation of two
independent standard normal r.v. (Tong, 1990, p. 11).
•
205
Theorem 4.190 — Distribution/moments of a linear transformation of a
multivariate normal distribution (Tong, 1990, p. 32, Theorem 3.3.3)
Let:
• X ∼ Normald (µ, Σ );
• C = [cij ] be any given k × d real matrix;
• b is any k × 1 real vector.
Then
Y = C X + b ∼ Normalk (C µ + b, C Σ C> ).
(4.134)
•
The family of multivariate normal distributions is closed not only under linear
transformations, as stated in the previous theorem, but also under linear combinations.
Corollary 4.191 — Distribution/moments of a linear combination of the
components of a multivariate normal random vector (Tong, 1990, p. 33, Corollary
3.3.3)
Let:
• X ∼ Normald (µ, Σ ) partitioned as in (4.126);
• C1 and C2 be two m × k and m × (d − k) real matrices, respectively.
Then
Y = C1 X 1 + C2 X 2 ∼ Normalm (µY , Σ Y ),
(4.135)
where the mean vector and the covariance matrix of Y are given by
µY
= C1 µ1 + C2 µ2
(4.136)
ΣY
>
>
>
= C1 Σ 11 C>
1 + C2 Σ 22 C1 + C1 Σ 12 C2 + C2 Σ 21 C1 ,
(4.137)
•
respectively.
206
The result that follows has already been proved in Chapter 3 and is a particular case
of Theorem 4.193.
Corollary 4.192 — Correlation and independence in a bivariate normal setting
(Tong, 1990, p. 8, Theorem 2.1.1)
•
Let X = (X1 , X2 ) ∼ Normal2 (µ, Σ ). Then X1 and X2 are independent iff ρ = 0.
In general, r.v. may be uncorrelated but highly dependent. But if a random vector
has a multivariate normal distribution then any two or more of its components that are
uncorrelated are independent.
Theorem 4.193 — Correlation and independence in a multivariate normal
setting (Tong, 1990, p. 31, Theorem 3.3.2)
Let X ∼ Normald (µ, Σ ) partitioned as in (4.126). Then X 1 and X 2 are independent
random vectors iff Σ 12 = Σ >
•
12 = 0k×(d−k) .
Corollary 4.194 — Linear combination of independent multivariate normal
random vectors (Tong, 1990, p. 33, Corollary 3.3.4)
Let X 1 , . . . , X N be independent Normald (µi , Σ i ), i = 1, . . . , N , random vectors. Then
!
N
N
N
X
X
X
Y =
ci µ i ,
c2i Σ i .
ci X i ∼ Normald
(4.138)
i=1
i=1
i=1
•
Proposition 4.195 — Independence between the sample mean vector and
covariance matrix (Tong, 1990, pp. 47–48)
Let:
• N be a positive integer;
• X 1 , . . . , X N be i.i.d. random vectors with a common Normald (µ, Σ ) distribution,
such that Σ is positive definite;
P
• X̄ N = N1 N
t=1 X t = (X̄1 , . . . , X̄d ) denote the sample mean vector, where
P
N
X̄i = N1 t=1 Xit and Xit the ith component of X t ;
• SN
=
[Sij ]i,j=1,...,d denote the
PN
1
Sij = N −1 t=1 (Xit − X̄i )(Xjt − X̄j ).
sample
covariance
matrix,
where
•
Then X̄ N and SN are independent.
207
Definition 4.196 — Mixed (central) moments
Let:
• X = (X1 , . . . , Xd ) be a random d−vector;
• r1 , . . . , rd ∈ IN .
Then:
• the mixed moment of order (r1 , . . . , rd ) of X is given by
E [X1r1 × . . . × Xdrd ] ,
(4.139)
P
and is also called a ( di=1 ri )th order moment of X;39
• the mixed central moment of order (r1 , . . . , rd ) of X is defined as
E {[X1 − E(X1 )]r1 × . . . [Xd − E(Xd )]rd } .
(4.140)
•
The Isserlis’ theorem is a formula that allows one to compute mixed moments of the
multivariate normal distribution with null mean vector40 in terms of the entries of its
covariance matrix.
Remarks 4.197 — Isserlis’ theorem (http://en.wikipedia.org/wiki/Isserlis’ theorem)
• In his original paper from 1918, Isserlis considered only the fourth-order moments,
in which case the formula takes appearance
E(X1 X2 X3 X4 ) = E(X1 X2 ) × E(X3 X4 ) + E(X1 X3 ) × E(X2 X4 )
+E(X1 X4 ) × E(X2 X3 ),
(4.141)
which can be written in terms of the covariances: σ12 × σ34 + σ13 × σ24 + σ14 × σ23 .
It also added that if (X1 , . . . , X2n ) is a zero mean multivariate normal random vector,
then
E(X1 . . . X2n−1 ) = 0
XY
E(X1 . . . X2n ) =
E(Xi Xj ),
39
(4.142)
(4.143)
See for instance http://en.wikipedia.org/wiki/Multivariate normal distribution.
Or mixed central moments of the difference between a multivariate normal random vector X and its
mean vector.
40
208
PQ
where the notation
means summing over all distinct ways of partitioning
X1 , . . . , X2n into pairs.
• This theorem is particularly important in particle physics, where it is known as
Wick’s theorem.
• Another applications include the analysis of portfolio returns, quantum field theory,
generation of colored noise, etc.
•
Theorem 4.198 — Isserlis’ theorem
(http://en.wikipedia.org/wiki/Multivariate normal distribution)
Let:
• X = (X1 , . . . , Xd ) ∼ Normald (µ, Σ );
i
hQ
d
ri
(X
−
µ
)
be the mixed central moment of order (r1 , . . . , rd ) of X;
• E
i
i
i=1
• k=
Pd
i=1 ri .
Then:
• if k is odd (i.e. k = 2n − 1, n ∈ IN )
"
#
d
Y
E
(Xi − µi )ri = 0;
(4.144)
i=1
• if k is even (i.e. k = 2n, n ∈ IN )
"
#
d
Y
XY
E
(Xi − µi )ri =
σij ,
(4.145)
i=1
PQ
where the
is taken over all allocations of the set {1, 2, . . . , n} into n (unordered)
pairs, that is, if you have a k = 2n = 6th order central moment, you will be summing
the products of n = 3 covariances.
•
209
Exercise 4.199 — Isserlis’ theorem
Let X = (X1 , . . . , X4 ) ∼ Normal4 (0, Σ = [σij ]). Prove that
(a) E(Xi4 ) = 3σii2 , i = 1, . . . , 4,
(b) E(Xi3 Xj ) = 3σii σij , i, j = 1, . . . , 4,
(c) E(Xi2 Xj2 ) = σii σjj + 2σij2 , i, j = 1, . . . , 4,
(d) E(Xi2 Xj Xl ) = σii σjl + 2σij σil , i, j, l = 1, . . . , 4,
(e) E(Xi Xj Xl Xn ) = σij σlm + σil σjn + σin σjl , i, j, l, n = 1, . . . , 4
(http://en.wikipedia.org/wiki/Isserlis’ theorem).
•
In passing from univariate to multivariate distributions, some essentially new features
require our attention: these features are connected not only with relations among sets
of variables and included covariance and correlation, but also regressions (conditional
expected values) and, generally, conditional distributions (Johnson and Kotz, 1969, p.
280).
Theorem 4.200 — Conditional distributions and regressions in the bivariate
normal setting (Tong, 1990, p. 8, Theorem 2.1.1)
Let X = (X1 , X2 ) be distributed according to a bivariate normal distribution with
parameters
# "
"
#
"
#
σ11 σ12
µ1
σ12
ρσ1 σ2
.
(4.146)
=
µ=
and Σ =
σ21 σ22
µ2
ρσ1 σ2
σ22
If |ρ| < 1 then
ρσ1
2
2
X1 |{X2 = x2 } ∼ Normal µ1 +
(x2 − µ2 ), σ1 (1 − ρ ) ,
σ2
(4.147)
−1
−1
X1 |{X2 = x2 } ∼ Normal µ1 + σ12 σ22
(x2 − µ2 ), σ11 − σ12 σ22
σ21 .
(4.148)
i.e.
•
Exercise 4.201 — Conditional distributions and regressions in the bivariate
normal setting
Prove that (4.147) and (4.148) are equivalent.
•
210
The following figure41 shows the conditional distribution of Y |{X = x0 } of a random
vector (X, Y ) with a bivariate normal distribution:
The inverse Mills ratio is the ratio of the probability density function over the
cumulative distribution function of a distribution and corresponds to a specific conditional
expectation, as stated below.
Definition 4.202 — Inverse Mills’ ratio
(http://en.wikipedia.org/wiki/Inverse Mills ratio; Tong, 1990, p. 174)
Let X = (X1 , X2 ) be a bivariate normal random vector with zero means, unit variances
and correlation coefficient ρ. Then the conditional expectation
E(X1 |{X2 > x2 }) = ρ
φ(x2 )
Φ(−x2 )
(4.149)
•
is often called the inverse Mills’ ratio.
Remark 4.203 — Inverse Mills’ ratio
(http://en.wikipedia.org/wiki/Inverse Mills ratio)
A common application of the inverse Mills’ ratio arises in regression analysis to take
account of a possible selection bias.
•
Exercise 4.204 — Conditional distributions and the inverse Mills’ ratio in the
bivariate normal setting
Assume that X1 represents the log-dose of insuline that has been administrated and X2
the decrease in blood sugar after a fixed amount of time. Also assume that (X1 , X2 ) has
a bivariate normal distribution with mean vector and covariance matrix
"
#
"
#
0.56
0.027
2.417
µ=
e Σ=
.
(4.150)
53
2.417 407.833
41
Once again taken from http://www.aiaccess.net/English/Glossaries/GlosMod/e gm multinormal distri.htm.
211
(a) Obtain the probability that the decrease in blood sugar exceeds 70, given that log-dose
of insuline that has been administrated is equal to 0.5.
(b) Determine the log-dose of insuline that has to be administrated so that the expected
value of the decrease in blood sugar equals 70.
(c) Obtain the expected value of the decrease in blood sugar, given that log-dose of
insuline that has been administrated exceeds 0.5.
•
Theorem 4.205 — Conditional distributions and regressions
multivariate normal setting (Tong, 1990, p. 35, Theorem 3.3.4)
Let X ∼ Normald (µ, Σ ) partitioned as in (4.126). Then
−1
(x
−
µ
),
Σ
−
Σ
Σ
Σ
X 1 |{X 2 = x2 } ∼ Normal µ1 + Σ 12 Σ −1
11
12 22
21 .
2
22
2
in
the
(4.151)
•
Exercise 4.206 — Conditional distributions and regressions in the multivariate
normal setting
Derive (4.148) from (4.151).
•
212
4.6.8
Multinomial distributions
The genesis and the definition of multinomial distributions follow.
Motivation 4.207 — Multinomial distribution
(http://en.wikipedia.org/wiki/Multinomial distribution)
The multinomial distribution is a generalization of the binomial distribution.
The binomial distribution is the probability distribution of the number of “successes”
in n independent Bernoulli trials, with the same probability of “success” on each trial.
In a multinomial distribution, the analog of the Bernoulli distribution is the categorical
distribution, where each trial results in exactly one of some fixed finite number d of possible
P
outcomes, with probabilities p1 , . . . , pd (pi ∈ [0, 1], i = 1, . . . , d, and di=1 pi = 1), and
there are n independent trials.
•
Definition 4.208 — Multinomial distribution (Johnson and Kotz, 1969, p. 281)
Consider a series of n independent trials, in each of which just one of d mutually exclusive
events E1 , . . . , Ed must be observed, and in which the probability of occurrence of event
P
Ei is equal to pi for each trial, with, of course, pi ∈ [0, 1], i = 1, . . . , d, and di=1 pi = 1.
Then the joint distribution of the r.v. N1 , . . . , Nd , representing the numbers of occurrences
of the events E1 , . . . , Ed (respectively) in n trials, is defined by
n!
P ({N1 = n1 , . . . , Nd = nd }) = Qd
i=1 ni !
×
d
Y
pni i ,
(4.152)
i=1
Pd
for ni ∈ IN0 , i = 1, . . . , d, such that i=1 ni = n. The random d−vector N = (N1 , . . . , Nd )
is said to have a multinomial distribution with parameters n and p = (p1 , . . . , pd )) — in
short N ∼ Multinomiald−1 (n, p = (p1 , . . . , pd )).42
•
Remark 4.209 — Special case of the multinomial distribution (Johnson and Kotz,
1969, p. 281)
Needless to say, we deal with the binomial distribution when d = 2, i.e.
d
Multinomial2−1 (n, p = (p, 1 − p)) = Binomial(n, p).
(4.153)
Curiously, J. Bernoulli, who worked with the binomial distribution, also used the
multinomial distribution.
•
42
The index d − 1 follows from the fact that the r.v. Nd (or any other component of N ) is redundant:
Pd−1
Nd = n − i=1 Ni .
213
Remark 4.210 — Applications of multinomial distribution
(http://controls.engin.umich.edu/wiki/index.php/Multinomial distributions)
Multinomial systems are a useful analysis tool when a “success-failure” description is
insufficient to understand the system. For instance, in chemical engineering applications,
multinomial distributions are relevant to situations where there are more than two possible
outcomes (temperature = high, med, low).
A continuous form of the multinomial distribution is the Dirichlet distribution
•
(http://en.wikipedia.org/wiki/Dirichlet distribution).43
Exercise 4.211 — Multinomial distribution (p.f.)
In a recent three-way election for a large country, candidate A received 20% of the votes,
candidate B received 30% of the votes, and candidate C received 50% of the votes.
If six voters are selected randomly, what is the probability that there will be exactly
one supporter for candidate A, two supporters for candidate B and three supporters for
candidate C in the sample? (http://en.wikipedia.org/wiki/Multinomial distribution) •
Exercise 4.212 — Multinomial distribution
A runaway reaction occurs when the heat generation from an exothermic reaction exceeds
the heat loss. Elevated temperature increases reaction rate, further increasing heat
generation and pressure buildup inside the reactor. Together, the uncontrolled escalation
of temperature and pressure inside a reactor may cause an explosion. The precursors to
a runaway reaction - high temperature and pressure - can be detected by the installation
of reliable temperature and pressure sensors inside the reactor. Runaway reactions can
be prevented by lowering the temperature and/or pressure inside the reactor before they
reach dangerous levels. This task can be accomplished by sending a cold inert stream into
the reactor or venting the reactor.
Les is a process engineer at the Miles Reactor Company that has been assigned to
work on a new reaction process. Using historical data from all the similar reactions
that have been run before, Les has estimated the probabilities of each outcome occurring
during the new process. The potential outcomes of the process include all permutations
of the possible reaction temperatures (low and high) and pressures (low and high). He
has combined this information into the table below:
43
The Dirichlet distribution is in turn the multivariate generalization of the beta distribution.
214
Outcome
Temperature
Pressure
Probability
1
high
high
0.013
2
high
low
0.267
3
low
high
0.031
4
low
low
0.689
Worried about risk of runaway reactions, the Miles Reactor Company is implementing
a new program to assess the safety of their reaction processes. The program consists
of running each reaction process 100 times over the next year and recording the reactor
conditions during the process every time. In order for the process to be considered safe,
the process outcomes must be within the following limits:
Outcome
Temperature
Pressure
Frequency
1
high
high
n1 = 0
2
high
low
n2 ≤ 20
3
low
high
n3 ≤ 2
4
low
low
n4 = 100 − n1 − n2 − n3
Help Les predict whether or not the new process is safe by
answering the following question:
“What is the probability that the
new process will meet the specifications of the new safety program?”
(http://controls.engin.umich.edu/wiki/index.php/Multinomial distributions).
•
Remark 4.213 — Multinomial expansion (Johnson and Kotz, 1969, p. 281)
If we recall the multinomial theorem44 then the expression of P ({N1 = n1 , . . . , Nd = nd })
Q
can be regarded as the coefficient of di=1 tni i in the multinomial expansion of
X
(t1 p1 + . . . + td pd )n =
P ({N1 = n1 , . . . , Nd = nd }) ×
d
Y
tni i ,
i=1
(n1 ,...,nd )
where the summation is taken over all (n1 , . . . , nd ) ∈ {(m1 , . . . , md ) ∈ IN0k :
and N ∼ Multinomiald−1 (n, p = (p1 , . . . , pd )).
44
(4.155)
Pd
i=1
mi = n}
•
For any positive integer d and any nonnegative integer n, we have
(x1 + . . . + xd )n =
n!
X
Qd
(n1 ,...,nd )
i=1 ni !
×
d
Y
xni i ,
(4.154)
i=1
where the summation is taken over all d−vectors of nonnegative integer indices n1 , . . . , nd such that the
sum of all ni is n. As with the binomial theorem, quantities of the form 00 which appear are taken to
equal 1. See http://en.wikipedia.org/wiki/Multinomial theorem for more details.
215
Definition 4.214 — Mixed factorial moments
Let:
• X = (X1 , . . . , Xd ) be a random d−vector;
• r1 , . . . , rd ∈ IN .
Then the mixed factorial moment of order (r1 , . . . , rd ) of X is equal to
h
i
(r )
(r )
E X1 1 × . . . × Xd d
= E {[X1 (X1 − 1) . . . (X1 − r1 + 1)] × . . . × [Xd (Xd − 1) . . . (Xd − rd + 1)]} . (4.156)
•
Marginal moments and marginal central moments, and covariances and correlations
between the components of a random vector can be written in terms of mixed
(central/factorial) moments. This is particularly useful when we are dealing with the
multinomial distribution.
Exercise 4.215 — Writing the variance and covariance in terms of mixed
(central/factorial) moments
Write
(a) the marginal variance of Xi and
(b) cov(Xi , Xj )
•
in terms of mixed factorial moments.
Proposition 4.216 — Mixed factorial moments of a multinomial distribution
(Johnson and Kotz, 1969, p. 284)
Let:
• N = (N1 , . . . , Nd ) ∼ Multinomiald−1 (n, p = (p1 , . . . , pd ));
• r1 , . . . , rd ∈ IN .
Then the mixed factorial moment of order (r1 , . . . , rd ) is equal to
d
h
i
Y
P
(r1 )
(rd )
( di=1 ri )
E N1 × . . . Nd
=n
×
pri i ,
(4.157)
i=1
Pd
where n(
i=1 ri )
=
(n−
n!
Pd
i=1 ri ))!
•
.
216
From the general formula (4.157), we find the expected value of Ni , and the covariances
and correlations between Ni and Nj .
Corollary 4.217 — Mean vector, covariance and correlation matrix
of a multinomial distribution
(Johnson and Kotz, 1969, p. 284;
http://en.wikipedia.org/wiki/Multinomial distribution)
The expected number of times the event Ei was observed over n trials, Ni , is
E(Ni ) = n pi ,
(4.158)
for i = 1, . . . , d.
The covariance matrix is as follows. Each diagonal entry is the variance
V (Ni ) = n pi (1 − pi ),
(4.159)
for i = 1, . . . , d. The off-diagonal entries are the covariances
cov(Ni , Nj ) = −n pi pj ,
(4.160)
for i, j = 1, . . . , d, i 6= j. All covariances are negative because, for fixed n, an increase in
one component of a multinomial vector requires a decrease in another component. The
covariance matrix is a d × d positive-semidefinite matrix of rank d − 1.
The off-diagonal entries of the corresponding correlation matrix are given by
r
pi pj
corr(Ni , Nj ) = −
,
(4.161)
(1 − pi ) (1 − pj )
for i, j = 1, . . . , d, i 6= j.45 Note that the number of trial drops out of this expression. •
Exercise 4.218 — Mean vector, covariance and correlation matrices of a
multinomial distribution
Use (4.157) to derive the entries of the mean vector, and the covariance and correlation
matrices of a multinomial distribution.
•
Exercise 4.219 — Mean vector and correlation matrix of a multinomial
distribution
Resume Exercise 4.211 and calculate the mean vector and the correlation matrix.
Comment the values you have obtained for the off-diagonal entries of the correlation
matrix.
•
45
The diagonal entries of the correlation matrix are obviously equal to 1.
217
Proposition 4.220 — Marginal distributions in a multinomial setting (Johnson
and Kotz, 1969, p. 281)
The marginal distribution of any Ni , i = 1, . . . , d, is Binomial with parameters n and pi .
I.e.
Ni ∼ Binomial(n, pi ).
(4.162)
•
for i = 1, . . . , d.
(4.162) is a special case of a more general result.
Proposition 4.221 — Joint distribution of a subset of r.v. from a multinomial
distribution (Johnson and Kotz, 1969, p. 281)
The joint distribution of any subset of s (s = 1, . . . , d − 1) r.v., say Na1 , . . . , Nas of the
P
Nj ’s, is also multinomial with an (s + 1)th r.v. equal to Nas+1 = n − si=1 Nai . In fact
P
P Na1 = na1 , . . . , Nas = nas , Nas+1 = n − si=1 nai P
P
Q
(4.163)
na
n− si=1 nai
n! P
= Qs na !×(n−
,
× si=1 pai i × (1 − si=1 pai )
s
na )!
i=1
i
i=1
i
for nai ∈ IN0 , i = 1, . . . , s such that
Pd
i=1
nai ≤ n.
•
Proposition 4.222 — Some regressions and conditional distributions in the
multinomial distribution setting (Johnson and Kotz, 1969, p. 284)
• The regression of Ni on Nj (j 6= i) is linear:
E(Ni |Nj ) = (n − Nj ) ×
pi
.
1 − pj
(4.164)
• The multiple regression of Ni on Nb1 , . . . , Nbr (bj 6= i, j = 1, . . . , r) is also linear:
E(Ni |{Nb1 , . . . , Nbr }) =
n−
r
X
!
Nbj
j=1
×
1−
p
Pri
j=1
pb j
.
(4.165)
• The random vector (Na1 , . . . Nas ) conditional on a event referring to any subset of the
remaining Nj ’s, say {Nb1 = nb1 , . . . , Nbr = nbr }, has also a multinomial distribution.
Its p.f. can be found in Johnson and Kotz (1969, p. 284).
•
218
Remark 4.223 — Conditional distributions and the simulation of a
multinomial distribution (Gentle, 1998, p. 106)
The following conditional distributions taken from Gentle (1998, p. 106) suggest a
procedure to generate pseudo-random vectors with a multinomial distribution:
• N1 ∼ Binomial(n, p1 );
p2
;
• N2 |{N1 = n1 } ∼ Binomial n − n1 , 1−p
1
• N3 |{N1 = n1 , N2 = n2 } ∼ Binomial n − n1 − n2 , 1−pp13−p2 ;
• ...
Pd−2
ni ,
• Nd−1 |{N1 = n1 , . . . , Nd−2 = nd−2 } ∼ Binomial n − i=1
d
• Nd |{N1 = n1 , . . . , Nd−1 = nd−1 } = n −
Pd−1
i=1
1−
pd−1
Pd−2
i=1
pi
;
ni .
Thus, we can generate a pseudo-random vector from a multinomial distribution
by sequentially generating independent pseudo-random numbers from the binomial
conditional distributions stated above.
Gentle (1998, p. 106) refers other ways of generating pseudo-random vectors from a
multinomial distribution.
•
Remark 4.224 — Speeding up the simulation of a multinomial distribution
(Gentle, 1998, p. 106)
To speed up the generation process, Gentle (1998, p. 106) recommends that we previously
order the probabilities p1 , . . . , pd in descending order — thus, getting the vector of
probabilities (p(d) , . . . , p(1) ), where p(d) = maxi=1,...,d pi , . . ., and p(1) = mini=1,...,d pi . Then
we generate d pseudo-random numbers with the following binomial distributions with
parameters
1. n and the largest probability of “success” p(d) , say n(d) ,
2. n − n(d) and
p(d−1)
,
1−p(d−1)
3. n − n(d) − n(d−1) and
say n(d−1) ,
p(d−2)
,
1−p(d−1) −p(d−2)
say n(d−1) ,
4. . . .
5. n −
Pd−2
i=1
n(d+1−i) and
p(2)
Pd−2
,
1− i=1
p(d+1−i)
say n(2) ,
219
and finally
6. assign n(1) = n −
Pd−1
i=1
•
n(d+1−i) .
Remark 4.225 — Speeding up the simulation of a multinomial distribution
(http://en.wikipedia.org/wiki/Multinomial distribution)
Assume the parameters p1 , . . . pd are already sorted descendingly (this is only to speed up
computation and not strictly necessary). Now, for each trial, generate a pseudo-random
number from U ∼ Uniform(0, 1), say u. The resulting outcome is the event Ej where
!
j0
X
k
pi ≥ u
j = arg min
0
j =1
=
i=1
FZ−1 (u),
(4.166)
with Z an integer r.v. that takes values 1, . . . , d, with probabilities p1 , . . . pd , respectively.
This is a sample for the multinomial distribution with n = 1.
The absolute frequencies of events E1 , . . . , Ed , resulting from n independent repetitions
of the procedure we just described, constitutes a pseudo-random vector from a multinomial
distribution with parameters n and p = (p1 , . . . pd ).
•
References
• Gentle, J.E. (1998). Random Number Generation and Monte Carlo Methods.
Springer-Verlag. (QA298.GEN.50103)
• Johnson, N.L. and Kotz, S. (1969). Discrete distributions John Wiley & Sons.
(QA273-280/1.JOH.36178)
• Karr, A.F. (1993). Probability. Springer-Verlag.
• Resnick, S.I. (1999). A Probability Path. Birkhäuser. (QA273.4-.67.RES.49925)
• Rohatgi, V.K. (1976). An Introduction to Probability Theory and Mathematical
Statistics. John Wiley & Sons. (QA273-280/4.ROH.34909)
• Tong, Y.L. (1990). The Multivariate Normal Distribution.
(QA278.5-.65.TON.39685)
Springer-Verlag.
• Walrand, J. (2004). Lecture Notes on Probability Theory and Random Processes.
Department of Electrical Engineering and Computer Sciences, University of
California, Berkeley.
220
Chapter 5
Convergence of sequences of random
variables
Throughout this chapter we assume that {X1 , X2 , . . .} is a sequence of r.v. and X is a
r.v., and all of them are defined on the same probability space (Ω, F, P ).
Stochastic convergence formalizes the idea that a sequence of r.v. sometimes is
expected to settle into a pattern.1 The pattern may for instance be that:
• there is a convergence of Xn (ω) in the classical sense to a fixed value X(ω), for each
and every event ω;
• the probability that the distance between Xn from a particular r.v. X exceeds any
prescribed positive value decreases and converges to zero;
• the series formed by calculating the expected value of the (absolute or quadratic)
distance between Xn and X converges to zero;
• the distribution of Xn may “grow” increasingly similar to the distribution of a
particular r.v. X.
Just as in analysis, we can distinguish among several types of convergence (Rohatgi,
1976, p. 240). Thus, in this chapter we investigate modes of convergence of sequences of
r.v.:
a.s.
• almost sure convergence ( →);
P
• convergence in probability (→);
1
See http://en.wikipedia.org/wiki/Convergence of random variables.
221
q.m.
• convergence in quadratic mean or in L2 ( → );
L1
• convergence in L1 or in mean (→);
d
• convergence in distribution (→).
It is important for the reader to be familiarized with all these modes of convergence,
the way they can be related and with the applications of such results and understand
their considerable significance in probability, statistics and stochastic processes.
5.1
Modes of convergence
∗
The first four modes of convergence (→, where ∗ = a.s., P, q.m., L1 ) pertain to the
d
sequence of r.v. and to X as functions of Ω, while the fifth (→) is related to the convergence
of d.f. (Karr, 1993, p. 135).
5.1.1
Convergence of r.v. as functions on Ω
Motivation 5.1 — Almost sure convergence (Karr, 1993, p. 135)
Almost sure convergence — or convergence with probability one — is the probabilistic
version of pointwise convergence known from elementary real analysis.
•
Definition 5.2 — Almost sure convergence (Karr, 1993, p. 135; Rohatgi, 1976, p.
249)
The sequence of r.v. {X1 , X2 , . . .} is said to converge almost surely to a r.v. X if
P
w : lim Xn (ω) = X(ω)
= 1.
(5.1)
n→+∞
a.s.
In this case we write Xn → X (or Xn → X with probability 1).
•
Remark 5.3 — Almost sure convergence
Equation (5.1) does not mean that limn→+∞ P ({w : Xn (ω) = X(ω)}) = 1.
•
Exercise 5.4 — Almost sure convergence
Let {X1 , X2 , . . .} be a sequence of independent r.v. such that Xn ∼ Bernoulli( n1 ), n ∈ IN .
a.s.
Prove that Xn 6 → 0, by deriving P ({Xn = 0, for every m ≤ n ≤ n0 }) and observing
that this probability does not converge to 1 as n0 → +∞ for all values of m (Rohatgi,
1976, p. 252, Example 9).
•
222
Motivation 5.5 — Convergence in probability (Karr, 1993, p. 135;
http://en.wikipedia.org/wiki/Convergence of random variables)
Convergence in probability essentially means that the probability that |Xn − X| exceeds
any prescribed, strictly positive value converges to zero.
The basic idea behind this type of convergence is that the probability of an “unusual”
outcome becomes smaller and smaller as the sequence progresses.
•
Definition 5.6 — Convergence in probability (Karr, 1993, p. 136; Rohatgi, 1976, p.
243)
The sequence of r.v. {X1 , X2 , . . .} is said to converge in probability to a r.v. X — denoted
P
by Xn → X — if
lim P ({|Xn − X| > }) = 0,
(5.2)
n→+∞
•
for every > 0.
Remarks 5.7 — Convergence in probability (Rohatgi,
http://en.wikipedia.org/wiki/Convergence of random variables)
1976,
p.
243;
• The definition of convergence in probability says nothing about the convergence
of r.v. Xn to r.v. X in the sense in which it is understood in real analysis. Thus,
P
Xn → X does not imply that, given > 0, we can find an N such that |Xn − X| < ,
for n ≥ N .
Definition 5.6 speaks only of the convergence of the sequence of probabilities
P (|Xn − X| > ) to zero.
• Formally, Definition 5.6 means that
∀, δ > 0, ∃Nδ : P ({|Xn − X| > }) < δ, ∀n ≥ Nδ .
(5.3)
• The concept of convergence in probability is used very often in statistics. For
example, an estimator is called consistent if it converges in probability to the
parameter being estimated.
• Convergence in probability is also the type of convergence established by the weak
law of large numbers.
•
223
Example 5.8 — Convergence in probability
Let {X1 , X2 , . . .} be a sequence of i.i.d. r.v. such that Uniform(0, θ), where θ > 0.
P
(a) Check if X(n) = maxi=1,...,n Xi → θ.
• R.v.
i.i.d.
Xi ∼ X, i ∈ IN
X ∼ Uniform(0, θ)
• D.f. of X

 0, x < 0
x
FX (x) =
, 0≤x≤θ
θ


1, x > θ
• New r.v.
X(n) = maxi=1,...,n Xi
• D.f. of X(n)
FX(n) (x) = [FX (x)]n


x<0
 0, x n
=
, 0≤x≤θ
θ


1,
x>θ
P
• Checking the convergence in probability X(n) → θ
Making use of the definition of this type of convergence and capitalizing on the
d.f. of X(n) , we get, for every > 0:
lim P |X(n) − θ| > = 1 − lim P θ − ≤ X(n) ≤ θ + n→+∞
n→+∞
h
i
= 1 − lim FX(n) (θ + ) − FXn) (θ − )
n→+∞

h
i


 1 − limn→+∞ FX(n) (θ) − FX(n) (θ − ) ,
=
0<<θ


 1 − lim
n→+∞ FX(n) (θ), ≥ θ

θ− n

 1 − limn→+∞ 1 − θ
=
= 1 − (1 − 0), 0 < < θ


1 − 1, ≥ θ
= 0.
224
• Conclusion
P
X(n) → θ.
Interestingly enough, X(n) not only is the MV estimator of θ but also a consistent
P
estimator of θ (X(n) → θ). However, E[X(n) ] = nθ/(n + 1) 6= θ, i.e. X(n) is a
biased estimator of θ.
P
(b) Prove that X(1:n) = mini=1,...,n Xi → 0.
• New r.v.
X(1:n) = mini=1,...,n Xi
• D.f. of X(1:n)
FX(1:n) (x) = 1 − [1 − FX (x)]n


x<0
 0,
x n
=
1− 1− θ , 0≤x≤θ


1,
x>θ
P
• Checking the convergence in probability X(1:n) → 0
For every > 0, we have
h
i
lim P |X(1:n) − 0| > = 1 − lim FX(1:n) () − FX(1:n) (−)
n→+∞
n→+∞
= 1 − lim FX(1:n) ()
n→+∞

n

 1 − limn→+∞ 1 − 1 − θ
=
= 1 − (1 − 0), 0 < < θ


1 − limn→+∞ FX(1:n) (θ) = 1 − 1, ≥ θ
= 0.
• Conclusion
P
X(1:n) → 0.
•
Remark 5.9 — Chebyshev(-Bienaymé)’s inequality and convergence in
probability
Chebyshev(-Bienaymé)’s inequality can be useful to prove that some sequences of r.v.
converge in probability to a degenerate r.v. (i.e. a constant).
•
225
Example 5.10 — Chebyshev(-Bienaymé)’s inequality and convergence in
probability
Let {X1 , X2 , . . .} be a sequence of r.v. such that Xn ∼ Gamma(n, n), n ∈ IN . Prove that
P
Xn → 1, by making use of Chebyshev(-Bienaymé)’s inequality.
• R.v.
Xn ∼ Gamma(n, n), n ∈ IN
E(Xn ) =
n
n
=1
V (Xn ) =
n
n2
=
1
n
P
• Checking the convergence in probability Xn → 1
The application of the definition of this type of convergence and Chebyshev(Bienaymé)’s inequality leads to
lim P (|Xn − 1| > ) =
n→+∞
≤
lim P
n→+∞
lim n→+∞
|Xn − E(Xn )| ≥ p
V (Xn )
!
p
V (Xn )
1
√
2
1
n
1
1
lim
2
n→+∞ n
= 0,
=
for every > 0.
• Conclusion
P
Xn → 1.
•
Exercise 5.11 — Chebyshev(-Bienaymé)’s inequality and convergence in
probability
Prove that X(n) = maxi=1,...,n Xi , where Xi ∼i.i.d. Uniform(0, θ), is a consistent estimator
n
θ
of θ > 0, by using Chebyshev(-Bienaymé)’s inequality and the fact that E[X(n) ] = n+1
n
2
and V [X(n) ] = (n+2)(n+1)2 θ .
•
226
Exercise 5.12 — Convergence in probability
Let {X1 , X2 , . . .} be a sequence of r.v. such that Xn ∼ Bernoulli( n1 ), n ∈ IN .
P
(a) Show that Xn → 0, by obtaining P ({|Xn | > }), for 0 < < 1 and ≥ 1 (Rohatgi,
1976, pp. 243–244, Example 5).
d
(b) Verify that E(Xnk ) → E(X k ), where k ∈ IN and X = 0.
•
Exercise 5.13 — Convergence in probability does not imply convergence of
kth. moments
d
Let {X1 , X2 , . . .} be a sequence of r.v. such that Xn = n × Bernoulli( n1 ), n ∈ IN , i.e.

1

 1 − n, x = 0
1
P ({Xn = x}) =
(5.4)
,
x=n
n


0,
otherwise.
P
Prove that Xn → 0, however E(Xnk ) 6→ E(X k ), where k ∈ IN and the r.v. X is degenerate
at 0 (Rohatgi, 1976, p. 247, Remark 3).
•
Motivation 5.14 — Convergence in quadratic mean and in L1
We have just seen that convergence in probability does not imply the convergence of
moments, namely of orders 2 or 1.
•
Definition 5.15 — Convergence in quadratic mean or in L2 (Karr, 1993, p. 136)
Let X, X1 , X2 , . . . belong to L2 . Then the sequence of r.v. {X1 , X2 , . . .} is said to converge
q.m.
L2
to X in quadratic mean (or in L2 ) — denoted by Xn → X (or Xn → X) — if
lim E (Xn − X)2 = 0.
n→+∞
(5.5)
•
Exercise 5.16 — Convergence in quadratic mean
Let {X1 , X2 , . . .} be a sequence of r.v. such that Xn ∼ Bernoulli n1 .
q.m.
Prove that Xn → X, where the r.v. X is degenerate at 0 (Rohatgi, 1976, p. 247,
Example 6).
•
Exercise 5.17 — Convergence in quadratic mean (bis)
1
Let {X1 , X2 , . . .} be a sequence of r.v. with P Xn = ± n1
= 2.
q.m.
Prove that Xn → X, where the r.v. X is degenerate at 0 (Rohatgi, 1976, p. 252,
Example 11).
•
227
Exercise 5.18 — Convergence in quadratic mean implies convergence of 2nd.
moments (Karr, 1993, p. 158, Exercise 5.6(a))
q.m.
Show that Xn → X ⇒ E(Xn2 ) → E(X 2 ) (Rohatgi, 1976, p. 248, proof of Theorem 8). •
Exercise 5.19 — Convergence in quadratic mean of partial sums (Karr, 1993, p.
159, Exercise 5.11)
Let X1 , X2 , . . . be pairwise uncorrelated r.v. with mean zero and partial sums
P
Sn = ni=1 Xi .
Prove that if there is a constant c such that V (Xi ) ≤ c, for every i, then
q.m.
Sn
•
→ 0 for all α > 21 .
nα
Definition 5.20 — Convergence in mean or in L1 (Karr, 1993, p. 136)
Let X, X1 , X2 , . . . belong to L1 . Then the sequence of r.v. {X1 , X2 , . . .} is said to converge
L1
to X in mean (or in L1 ) — denoted by Xn → X — if
lim E (|Xn − X|) = 0.
(5.6)
n→+∞
•
Exercise 5.21 — Convergence in mean implies convergence of 1st. moments
(Karr, 1993, p. 158, Exercise 5.6(b))
L1
Prove that Xn → X ⇒ E(Xn ) → E(X).
•
228
5.1.2
Convergence in distribution
Motivation 5.22 — Convergence in distribution
(http://en.wikipedia.org/wiki/Convergence of random variables)
Convergence in distribution is very frequently used in practice, most often it arises from
the application of the central limit theorem.
•
Definition 5.23 — Convergence in distribution (Karr, 1993, p. 136; Rohatgi, 1976,
pp. 240–1)
d
The sequence of r.v. {X1 , X2 , . . .} converges to X in distribution — denoted by Xn → X
— if
lim FXn (x) = FX (x),
(5.7)
n→+∞
•
for all x at which FX is continuous.
Remarks 5.24 — Convergence in distribution
(http://en.wikipedia.org/wiki/Convergence of random variables; Karr, 1993, p. 136;
Rohatgi, 1976, p. 242)
• With this mode of convergence, we increasingly expect to see the next r.v. in a
sequence of r.v. becoming better and better modeled by a given d.f., as seen in
exercises 5.25 and 5.26.
• It must be noted that it is quite possible for a given sequence of d.f. to converge to
a function that is not a d.f., as shown in Example 5.27 and Exercise 5.28.
• The requirement that only the continuity points of FX should be considered is
essential, as we shall see in exercises 5.29 and 5.30.
• The convergence in distribution does not imply the convergence of corresponding
p.(d.)f., as shown in Exercise 5.32. Sequences of absolutely continuous r.v. that
converge in distribution to discrete r.v. (and vice-versa) are obvious illustrations, as
shown in examples 5.31 and 5.33.
•
Exercise 5.25 — Convergence in distribution
Let X1 , X2 , . . . , Xn be i.i.d. r.v. with common p.d.f.
(
1
, 0<x<θ
θ
f (x) =
0, otherwise,
229
(5.8)
where 0 < θ < +∞, and X(n) = max1,...,n Xi .
d
Show that X(n) → θ (Rohatgi, 1976, p. 241, Example 2).
•
Exercise 5.26 — Convergence in distribution (bis)
Let:
• {X1 , X2 , . . .} be a sequence of r.v. such that Xn ∼ Bernoulli(pn ), n = 1, 2, . . .;
• X ∼ Bernoulli(p).
d
Prove that Xn → X iff pn → p.
•
Example 5.27 — A sequence of d.f. converging to a non d.f. (Murteira, 1979, pp.
330–331)
Let {X1 , X2 , . . .} be a sequence of r.v. with d.f.


x < −n
 0,
x+n
(5.9)
FXn (x) =
, −n ≤ x < n
2n


1,
x ≥ n.
Please note that limn→+∞ FXn (x) = 21 , x ∈ IR, as suggested by the graph below with some
terms of the sequence of d.f., for n = 1, 103 , 106 (from top to bottom):
1.0
0.8
0.6
0.4
0.2
-1000
500
-500
1000
Consequently, the limit of the sequence of d.f. is not itself a d.f.
Exercise 5.28 — A sequence of d.f. converging to a non d.f.
Consider the sequence of d.f.
(
0, x < n
FXn (x) =
1, x ≥ n,
•
(5.10)
where FXn (x) is the d.f. of the r.v. Xn degenerate at x = n.
Verify that FXn (x) converges to a function (that is identically equal to 0!!!) which is
not a d.f. (Rohatgi, 1976, p. 241, Example 1).
•
230
Exercise 5.29 — The requirement that only the continuity points of FX should
be considered is essential
Let Xn ∼ Uniform 21 − n1 , 12 + n1 and X be a r.v. degenerate at 12 .
d
(a) Prove that Xn → X (Karr, 1993, p. 142).
(b) Verify that FXn 12 = 12 for each n, and these values do not converge to FX 12 = 1.
Is there any contradiction with the convergence in distribution previously proved?
(Karr, 1993, p. 142.)
•
Exercise 5.30 — The requirement that only the continuity points of FX should
be considered is essential (bis)
Let Xn ∼ Uniform 0, n1 and X a r.v. degenerate at 0.
d
Prove that Xn → X, even though FXn (0) = 0, for all n, and FX (0) = 1,
that is, the convergence of d.f. fails at the point x = 0 where FX is discontinuous
(http://en.wikipedia.org/wiki/Convergence of random variables).
•
Example 5.31 — Convergence in distribution does not imply convergence of
corresponding p.(d.)f. (Murteira, 1979, p. 331)
Let {X1 , X2 , . . .} be a sequence of r.v. such that Xn ∼ Normal 0, n12 .
An analysis of the representation of some terms of the sequence of d.f. (e.g. n =
1, 10, 50, from left to right in the following graph) and the notion of convergence in
d
d
distribution leads us to conclude that Xn → X, where X = 0, even though


0−0
1
lim FXn (0) = lim Φ  q  = Φ(0) =
n→+∞
n→
2
1
n2
1.0
0.8
0.6
0.4
0.2
-3
-2
1
-1
231
2
3
and


 0, x < 0
1
lim FXn (x) =
, x=0
2
n→+∞


1, x > 0
is not a a d.f. (it is not left- or right-continuous).
d
Note that X = 0, therefore the limit d.f. of the sequence of r.v. {X1 , X2 , . . .} is the
Heaviside function, i.e. FX (x) = I[0,+∞) (x).
•
Exercise 5.32 — Convergence in distribution does not imply convergence of
corresponding p.(d.)f.
Let {X1 , X2 , . . .} be a sequence of r.v. with p.f.
(
1, x = 2 + n1
(5.11)
P ({Xn = x}) =
0, otherwise.
d
(a) Prove that Xn → X, where X a r.v. degenerate at 2.
(b) Verify that none of the p.f. P ({Xn = x}) assigns any probability to the point x = 2,
for all n, and that P ({Xn = x}) → 0 for all x (Rohatgi, 1976, p. 242, Example 4). •
Example 5.33 — A sequence of discrete r.v. that converges in distribution to
an absolutely continuous r.v. (Rohatgi, 1976, p. 256, Exercise 10)
Let:
• {X1 , X2 , . . .} be a sequence of r.v. such that Xn ∼ Geometric nλ , where n > λ > 0;
• {Yn , n ∈ IN } a sequence of r.v. such taht Yn =
Xn
.
n
d
Show that Yn → Exponential (λ).
• R.v.
Xn ∼ Geometric
λ
n
, n ∈ IN
• P.f. of Xn and Yn
λ x−1
n
× nλ , x = 1, 2, . . .
ny λ
× n , y = n1 , n2 , . . .
P (Yn = y) = P (Xn = ny) = 1 − nλ
P (Xn = x) = 1 −
232
• D.f. of Yn
FYn (y) = FXn (ny)
(
0,
=
P[ny]
y < n1
1
x=1 P (Xn = x), y ≥ n ,
where [ny] represents the integer part of the real number ny and
[ny]−1 [ny]
X
x
λ
λ
P (Xn = x) =
1−
×
n
n
x=1
i=0
[ny]
λ
.
= 1− 1−
n
X
• Checking the convergence in distribution
Let us remind the reader that [ny] = ny − , for some ∈ [0, 1). Thus:
lim FYn (y) =
n→+∞
=
=
=
[ny]
λ
1 − lim 1 −
n→+∞
n
ny
−
λ
λ
× lim 1 −
1 − lim 1 −
n→+∞
n→+∞
n
n
n y
λ
1 − lim 1 −
×1
n→+∞
n
1 − e−λy
= FExponential(λ) (y).
• Conclusion
d
Yn → Exponential(λ).
•
Exercise 5.34 — A sequence of discrete r.v. that converges in distribution to
an absolutely continuous r.v. (bis)
Let {X1 , X2 , . . .} be a sequence of discrete r.v. such that Xn ∼ Uniform{0, 1, . . . , n}.
d
•
Prove that Yn = Xnn → Uniform(0, 1).2
2
This result is very important in the generation of pseudo-random numbers from the Uniform(0, 1)
distribution by using computers since these machines “deal” with discrete mathematics.
233
The following table condenses the definitions of convergence of sequences of r.v.
Mode of convergence
a.s.
Xn → X (almost sure)
P
Xn → X (in probability)
q.m
Xn → X (in quadratic mean)
L1
Xn → X (in L1 )
d
Xn → X (in distribution)
Assumption
Defining condition
—
P ({w : Xn (ω) → X(ω)}) = 1
—
X, X1 , X2 , . . . ∈ L2
P ({|Xn − X| > }) → 0, for all > 0
E (Xn − X)2 → 0
X, X1 , X2 , . . . ∈ L1
E (|Xn − X|) → 0
—
FXn (x) → FX (x), at continuity points x of FX
Exercise 5.35 — Modes of convergence and uniqueness of limit (Karr, 1993, p.
158, Exercise 5.1)
Prove that for all five forms of convergence the limit is unique. In particular:
∗
∗
d
d
a.s.
(a) if Xn → X and Xn → Y , where ∗ = a.s., P, q.m., L1 , then X → Y ;
d
(b) if Xn → X and Xn → Y , then X → Y ;
•
Exercise 5.36 — Modes of convergence and the vector space structure of the
family of r.v. (Karr, 1993, p. 158, Exercise 5.2)
Prove that, for ∗ = a.s., P, q.m., L1 ,
∗
∗
Xn → X ⇔ Xn − X → 0,
(5.12)
i.e. the four function-based forms of convergence are compatible with the vector space
structure of the family of r.v.
•
234
5.1.3
Alternative criteria
The definition of almost sure convergence and its verification are far from trivial. More
tractable criteria have to be stated...
Proposition 5.37 — Relating almost sure convergence and convergence in
probability (Karr, 1993, p. 137; Rohatgi, 1976, p. 249)
a.s.
Xn → X iff
∀ > 0, lim P
sup |Xk − X| > = 0,
(5.13)
n→+∞
k≥n
i.e.
a.s.
Xn → X
P
⇔ Yn = sup |Xk − X| → 0.
(5.14)
k≥n
•
Remarks 5.38 — Relating almost sure convergence and convergence in
probability (Karr, 1993, p. 137; Rohatgi, 1976, p. 250, Remark 6)
• Proposition 5.37 states an equivalent form of almost sure convergence that
illuminates its relationship to convergence in probability.
a.s.
• Xn → 0 means that,
∀, η > 0, ∃n0 ∈ IN : P
sup |Xk | > < η.
(5.15)
k≥n0
Indeed, we can write, equivalently, that
!
lim P
n→+∞
[
{|Xk | > }
= 0,
(5.16)
k≥n
•
for > 0 arbitrary.
Exercise 5.39 — Relating almost sure convergence and convergence in
probability
Prove Proposition 5.37 (Karr, 1993, p. 137; Rohatgi, 1976, p. 250).
•
235
Exercise 5.40 — Relating almost sure convergence and convergence in
probability (bis)
1
= 2.
Let {X1 , X2 , . . .} be a sequence of r.v. with P Xn = ± n1
a.s.
Prove that Xn → X, where the r.v. X is degenerate at 0, by using (5.16) (Rohatgi,
1976, p. 252).
•
Theorem 5.41 — Cauchy criterion (Rohatgi, 1976, p. 270)
a.s.
Xn → X ⇔
lim P
sup |Xn+m − Xn | ≤ = 1, ∀ > 0.
n→+∞
(5.17)
m
•
Exercise 5.42 — Cauchy criterion
Prove Theorem 5.41 (Rohatgi, 1976, pp. 270–2).
•
Definition 5.43 — Complete convergence (Karr, 1993, p. 138)
The sequence of r.v. {X1 , X2 , . . .} is said to converge completely to X if
+∞
X
P ({|Xn − X| > } ) < +∞,
(5.18)
n=1
•
for every > 0.
The next results relate complete convergence, which is stronger than almost sure
convergence, and sometimes more convenient to establish (Karr, 1993, p. 137).
Proposition 5.44 — Relating
convergence (Karr, 1993, p. 138)
+∞
X
almost
sure
convergence
a.s.
P ({|Xn − X| > } ) < +∞, ∀ > 0 ⇒ Xn → X.
and
complete
(5.19)
n=1
•
Remark 5.45 — Relating almost sure convergence and complete convergence
(Karr, 1993, p. 138)
P
a.s.
Xn → X iff the probabilities P ({|Xn − X| > } ) converge to zero, while Xn → X if (but
not only if) the convergence of probabilities P ({|Xn − X| > } ) is fast enough that their
P
sum, +∞
•
n=1 P ({|Xn − X| > } ), is finite.
236
Exercise 5.46 — Relating almost sure convergence and complete convergence
Show Proposition 5.44, by using the (1st.) Borel-Cantelli lemma (Karr, 1993, p. 138). •
Theorem 5.47 — Almost sure convergence of a sequence of independent r.v.
(Rohatgi, 1976, p. 265)
Let {X1 , X2 , . . .} be a sequence of independent r.v. Then
a.s.
Xn → 0 ⇔
+∞
X
P ({|Xn | > } ) < +∞, ∀ > 0.
(5.20)
n=1
•
Exercise 5.48 — Almost sure convergence of a sequence of independent r.v.
Prove Theorem 5.47 (Rohatgi, 1976, pp. 265–6).
•
The definition of convergence in distribution is cumbersome because of the proviso
regarding continuity points of the limit d.f. FX . An alternative criterion follows.
Theorem 5.49 — Alternative criterion for convergence in distribution (Karr,
1993, p. 138)
Let C be the set of bounded, continuous functions f : IR → IR. Then
d
Xn → X
⇔ E[f (Xn )] → E[f (X)], ∀ f ∈ C.
(5.21)
•
Remark 5.50 — Alternative criterion for convergence in distribution (Karr,
1993, p. 138)
Theorem 5.49 provides a criterion for convergence in distribution which is superior to
the definition of convergence in distribution in that one need not to deal with continuity
points of the limit d.f.
•
Exercise 5.51 — Alternative criterion for convergence in distribution
Prove Theorem 5.49 (Karr, 1993, pp. 138–139).
•
Since in the proof of Theorem 5.49 the continuous functions used to approximate
indicator functions can be taken to be arbitrarily smooth we can add a sufficient condition
that guarantees convergence in distribution.
237
Corollary 5.52 — Sufficient condition for convergence in distribution (Karr,
1993, p. 139)
Let:
• k be a fixed non-negative integer;
• C(k) be the space of bounded, k−times uniformly continuously differentiable
functions f : IR → IR.
Then
d
E[f (Xn )] → E[f (X)], ∀ f ∈ C(k) ⇒ Xn → X.
(5.22)
•
The next table summarizes the alternative criteria and sufficient conditions for almost
sure convergence and convergence in distribution of sequences of r.v.
Alternative criterion or sufficient condition
∀ > 0, limn→+∞ P supk≥n |Xk − X| > = 0
Mode of convergence
a.s.
⇔
Xn → X
Yn = supk≥n |Xk − X| → 0
⇔
Xn → X
limn→+∞ P ({supm |Xn+m − Xn | ≤ }) = 1, ∀ > 0
P+∞
n=1 P ({|Xn − X| > } ) < +∞, ∀ > 0
⇔
Xn → X
⇒
Xn → X
E[f (Xn )] → E[f (X)], ∀ f ∈ C
⇔
Xn → X
E[f (Xn )] → E[f (X)], ∀ f ∈ C(k) for a fixed k ∈ IN0
⇒
Xn → X
P
a.s.
a.s.
a.s.
d
d
P
We should also add that Grimmett and Stirzaker (2001, p. 310) state that if Xn → X
Lr
and P ({|Xn | ≤ k}) = 1, for all n and some k, then Xn → X, for all r ≥ 1,3 namely
q.m.
L1
Xn → X (which in turn implies Xn → X).
3
Let X, X1 , X2 , . . . belong to Lr (r ≥ 1). Then the sequence of r.v. {X1 , X2 , . . .} is said to converge
Lr
to X in Lr ) — denoted by Xn → X — if limn→+∞ E (|Xn − X|r ) = 0 (Grimmett and Stirzaker, 2001,
p. 308).
238
5.2
Relationships among the modes of convergence
Given the plethora of modes of convergence, it is natural to inquire how they can be
always related or hold true in the presence of additional assumptions (Karr, 1993, pp. 140
and 142).
5.2.1
Implications always valid
Proposition 5.53 — Almost sure convergence
probability (Karr, 1993, p. 140; Rohatgi, 1976, p. 250)
a.s.
Xn → X
implies
convergence
P
⇒ Xn → X.
in
(5.23)
•
Exercise 5.54 — Almost sure convergence implies convergence in probability
Show Proposition 5.53 (Karr, 1993, p. 140; Rohatgi, 1976, p. 251).
•
Proposition 5.55 — Convergence in quadratic mean implies convergence in L1
(Karr, 1993, p. 140)
q.m.
Xn → X
L1
⇒ Xn → X.
(5.24)
•
Exercise 5.56 — Convergence in quadratic mean implies convergence in L1
Prove Proposition 5.55, by applying Cauchy-Schwarz’s inequality (Karr, 1993, p. 140). •
Proposition 5.57 — Convergence in L1 implies convergence in probability
(Karr, 1993, p. 141)
L1
Xn → X
P
⇒ Xn → X.
(5.25)
•
Exercise 5.58 — Convergence in L1 implies convergence in probability
Prove Proposition 5.57, by using Chebyshev’s inequality (Karr, 1993, p. 141).
•
Proposition 5.59 — Convergence in probability implies convergence in
distribution (Karr, 1993, p. 141)
P
Xn → X
d
⇒ Xn → X.
(5.26)
•
239
Exercise 5.60 — Convergence in probability
distribution
Show Proposition 5.59, (Karr, 1993, p. 141).
implies
convergence
in
•
Figure 5.1 shows that convergence in distribution is the weakest form of convergence,
since it is implied by all other types of convergence studied so far.
q.m.
Xn → X
⇒
L1
Xn → X
⇒
P
Xn → X
d
⇒ Xn → X
⇒
a.s.
Xn → X
Figure 5.1: Implications always valid between modes of convergence.
Grimmett and Stirzaker (2001, p. 314) refer that convergence in distribution is the
weakest form of convergence for two reasons: it only involves d.f. and makes no reference
to an underlying probability space.4 However, convergence in distribution has an useful
representation in terms of almost sure convergence, as stated in the next theorem.
Theorem 5.61 — Skorokhod’s representation theorem (Grimmett and Stirzaker,
2001, p. 314)
Let {X1 , X2 , . . .} be a sequence of r.v., {F1 , F2 , . . .} the associated sequence of d.f. and X
d
be a r.v. with d.f. F . If Xn → X then there is a probability space (Ω0 , F 0 , P 0 ) and r.v.
{Y1 , Y2 , . . .} and Y mapping Ω0 into IR such that {Y1 , Y2 , . . .} and Y have d.f. {F1 , F2 , . . .}
d
and F and Yn → Y .
•
Remark 5.62 — Skorokhod’s representation theorem (Grimmett and Stirzaker,
2001, p. 315)
Although Xn may fail to converge to X in any mode than in distribution, there is a
sequence of r.v. {Y1 , Y2 , . . .} such that Yn is identically distributed to Xn , for every n,
which converges almost surely to a “copy” of X.
•
4
Let us remind the reader that that there is an equivalent formulation of convergence in distribution
which involves d.f. alone: the sequence of d.f. {F1 , F2 , . . .} converges to the d.f. F , if limn→+∞ Fn (x) =
F (x) at each point x where F is continuous (Grimmett and Stirzaker, 2001, p. 190).
240
5.2.2
Counterexamples
Counterexamples to all implications among the modes of convergence (and more!) are
condensed in Figure 5.2 and presented by means of several exercises.
q.m.
Xn → X
6⇐
L1
Xn → X
6⇐
P
d
Xn → X
6⇐ Xn → X
6⇐
a.s.
Xn → X
Figure 5.2: Counterexamples to implications among the modes of convergence.
Before proceeding with exercises, recall exercises 5.4 and 5.12 which pertain to the
sequence of r.v. {X1 , X2 , . . .}, where Xn ∼ Bernoulli( n1 ), n ∈ IN . In the first exercise
a.s.
P
we proved that Xn 6 → 0, whereas in the second one we concluded that Xn → 0. Thus,
P
a.s.
combining these results we can state that Xn → 0 6⇒ Xn → 0.
Exercise 5.63 — Almost sure convergence does not imply convergence in
quadratic mean
Let {X1 , X2 , . . .} be a sequence of r.v. such that

1

 1 − n, x = 0
1
P ({Xn = x}) =
(5.27)
,
x=n
n


0,
otherwise.
a.s.
P
d
L1
q.m.
6 0 and Xn 6 → 0
Prove that Xn → 0, and, hence, Xn → 0 and Xn → 0, but Xn →
(Karr, 1993, p. 141, Counterexample a)).
•
Exercise 5.64 — Almost sure convergence does not imply convergence in
quadratic mean (bis)
Let {X1 , X2 , . . .} be a sequence of r.v. such that

1

 1 − nr , x = 0
1
P ({Xn = x}) =
(5.28)
,
x=n
nr


0,
otherwise,
where r ≥ 2.
q.m.
a.s.
Prove that Xn → 0, but Xn 6 → 0 for r = 2 (Rohatgi, 1976, p. 252, Example 10).
241
•
Exercise 5.65 — Convergence in quadratic mean does not imply almost sure
convergence
Let Xn ∼ Bernoulli n1 .
q.m.
a.s.
Prove that Xn → 0, but Xn 6 → 0 (Rohatgi, 1976, p. 252, Example 9).
•
Exercise 5.66 — Convergence in L1 does not imply convergence in quadratic
mean
Let {X1 , X2 , . . .} be a sequence of r.v. such that

1

0
 1 − n, x = √
1
(5.29)
P ({Xn = x}) =
,
x= n
n


0,
otherwise.
L1
a.s.
q.m.
Show that Xn → 0 and Xn → 0, however Xn 6 → 0 (Karr, 1993, p. 141,
Counterexample b)).
•
Exercise 5.67 — Convergence in probability does not imply almost sure
convergence
For each positive integer n there exists integers m and k (uniquely determined) such that
n = 2k + m, m = 0, 1, . . . , 2k − 1, k = 0, 1, 2, . . .
Thus, for n = 1, k = m = 0; for n = 5, k = 2, m = 1; and so on.
Define r.v. Xn , for n = 1, 2, . . ., on Ω = [0, 1] by
(
2k , 2mk ≤ w < m+1
2k
Xn (ω) =
0, otherwise.
(5.30)
(5.31)
Let the probability distribution of Xn be given by P ({I}) = length of the interval I ⊂ Ω.
Thus,

1

 1 − 2k , x = 0
1
(5.32)
P ({Xn = x}) =
,
x = 2k
2k


0,
otherwise.
P
a.s.
Prove that Xn → 0, but Xn 6 → 0 (Rohatgi, 1976, pp. 251–2, Example 8).
242
•
Exercise 5.68 — Convergence in distribution does not imply convergence in
probability
Let {X2 , X3 , . . .} be a sequence of r.v. such that


x<0
 0,
1
1
FXn (x) =
(5.33)
− n, 0 ≤ x < 1
2


1,
x ≥ 1,
i.e. Xn ∼ Bernoulli 21 + n1 , n = 2, 3, . . .
d
Prove that Xn → X, where X ∼ Bernoulli 12 and independent of any Xn , but
P
Xn →
6 X (Karr, 1993, p. 142, Counterexample d)).
•
Exercise 5.69 — Convergence in distribution does not imply convergence in
probability (bis)
Let X, X1 , X2 , . . . be i.i.d. r.v. and let the joint p.f. of (X, Xn ) be P ({X = 0, Xn = 1}) =
P ({X = 1, Xn = 0}) = 12 .
d
P
Prove that Xn → X, but Xn → X (Rohatgi, 1976, p. 247, Remark 2).
243
•
5.2.3
Implications of restricted validity
Proposition 5.70 — Convergence in distribution to a constant implies
convergence in probability (Karr, 1993, p. 140; Rohatgi, 1976, p. 246)
Let {X1 , X2 , . . .} be a sequence of r.v. and c ∈ IR. Then
d
P
Xn → c ⇒ Xn → c.
(5.34)
•
Remark 5.71 — Convergence in distribution to a constant is equivalent to
convergence in probability (Rohatgi, 1976, p. 246)
P
d
If we add to the previous result the fact that Xn → c ⇒ Xn → c, we can conclude that
P
d
Xn → c ⇔ Xn → c.
(5.35)
•
Exercise 5.72 — Convergence in distribution to a constant implies convergence
in probability
Show Proposition 5.70 (Karr, 1993, p. 140).
•
Exercise 5.73 — Convergence in distribution to a constant implies convergence
in probability (bis)
Let (X1 , . . . , Xn ) be a random vector where Xi are i.i.d. r.v. with common p.d.f.
fX (x) = θx−2 × I[θ,+∞) (x),
where θ ∈ IR+ .
(a) After having proved that
FX(1:n) (x) = P
min Xi ≤ x = [1 − (θ/x)n ] × I[θ,+∞) (x),
i=1,...,n
(5.36)
d
derive the following result: X(1:n) → θ.
•
(b) Is X(1:n) a consistent estimator of θ?
Definition 5.74 — Uniform integrability (Karr, 1993, p. 142)
A sequence of r.v. {X1 , X2 , . . .} is uniformly integrable if Xn ∈ L1 for each n ∈ IN and if
lim sup E(|Xn |; {|Xn | > a}) = 0.
a→+∞
(5.37)
n
Recall that the expected value of a r.v. X over an event A is given by E(X; A) = E(X ×
1A ).
•
244
Proposition 5.75 — Alternative criterion for uniform integrability (Karr, 1993,
p. 143)
A sequence of r.v. {X1 , X2 , . . .} is uniformly integrable iff
• supn E(|Xn |) < +∞ and
• {X1 , X2 , . . .} is uniformly absolutely continuous: for each > 0 there is δ > 0 such
that supn E(|Xn |; A) < whenever P (A) > δ.
•
Proposition 5.76 — Combining convergence in probability and uniform
integrability is equivalent to convergence in L1 (Karr, 1993, p. 144)
Let X, X1 , X2 , . . . ∈ L1 . Then
L1
P
Xn → X and {X1 , X2 , . . .} is uniformly integrable ⇔ Xn → X.
(5.38)
•
Exercise 5.77 — Combining convergence in
integrability is equivalent to convergence in L1
Prove Proposition 5.76 (Karr, 1993, p. 144).
probability
and
uniform
•
Exercise 5.78 — Combining convergence in probability of the sequence of r.v.
and convergence of sequence of the means implies convergence in L1 (Karr,
1993, p. 160, Exercise 5.16)
Let X, X1 , X2 , . . . be positive r.v.
L1
P
•
Prove that if Xn → X and E(Xn ) → E(X), then Xn → X.
Exercise 5.79 — Increasing character and convergence in probability
combined imply almost sure convergence (Karr, 1993, p. 160, Exercise 5.15)
P
a.s.
Show that if X1 ≤ X2 ≤ . . . and Xn → X, then Xn → X.
•
Exercise 5.80 — Strictly decreasing and positive character and convergence
in probability combined imply almost sure convergence (Rohatgi, 1976, p. 252,
Theorem 13)
Let {X1 , X2 , . . .} be a strictly decreasing sequence of positive r.v.
P
a.s.
Prove that if Xn → 0 then Xn → 0.
•
245
5.3
Convergence under transformations
Since the original sequence(s) of r.v. is (are) bound to be transformed, it is natural to
inquire whether the modes of convergence are preserved under continuous mappings and
algebraic operations of the r.v.
5.3.1
Continuous mappings
Only convergence almost surely, in probability and in distribution are preserved under
continuous mappings (Karr, 1993, p. 145).
Theorem 5.81 — Preservation of {a.s., P, d}−convergence under continuous
mappings (Karr, 1993, p. 148)
Let:
• {X1 , X2 , . . .} be a sequence of r.v. and X a r.v.;
• g : IR → IR be a continuous function.
Then
∗
∗
Xn → X
⇒ g(Xn ) → g(X), ∗ = a.s., P, d.
(5.39)
•
Exercise 5.82 — Preservation of {a.s., P, d}−convergence under continuous
mappings
Show Theorem 5.81 (Karr, 1993, p. 148).
•
5.3.2
Algebraic operations
With the exception of the convergence in distribution, addition is preserved by the modes
of convergence of r.v. as functions on Ω, as stated in the next theorem.
Theorem 5.83 — Preservation of {a.s., P, q.m, L1 }−convergence under addition
(Karr, 1993, p. 145)
∗
∗
Let Xn → X and Yn → Y , where ∗ = a.s., P, q.m, L1 . Then
∗
Xn + Yn → X + Y, ∗ = a.s., P, q.m, L1 .
(5.40)
•
246
Remark 5.84 — Preservation of {a.s., P, q.m, L1 }−convergence under addition
Under the conditions of Theorem 5.83,
∗
• Xn ± Yn → X ± Y, ∗ = a.s., P, q.m, L1 .
•
Exercise 5.85 — Preservation of {a.s., P, q.m, L1 }−convergence under addition
Prove Theorem 5.83 (Karr, 1993, pp. 145–6).
•
Convergence in distribution is only preserved under addition if one of the limits is
constant.
Theorem 5.86 — Slutsky’s theorem or preservation of d−convergence under
(restricted) addition (Karr, 1993, p. 146)
Let:
d
• Xn → X;
d
• Yn → c, c ∈ IR.
Then
d
Xn + Yn → X + c.
(5.41)
•
Remarks 5.87 — Slutsky’s theorem or preservation of d−convergence under
(restricted) addition and subtraction
(http://en.wikipedia.org/wiki/Slutsky’s theorem; Rohatgi, 1976, p. 253)
• The requirement that {Yn } converges in distribution to a constant is important —
if it were to converge to a non-degenerate random variable, Theorem 5.86 would be
no longer valid.
• Theorem 5.86 remains valid if we replace all convergences in distribution with
convergences in probability because it implies the convergence in distribution.
• Moreover, Theorem 15 (Rohatgi, 1976, p. 253) reads as follows:
d
P
d
Xn → X, Yn → c, c ∈ IR ⇒ Xn ± Yn → X ± c.
d
(5.42)
In this statement, the condition of Yn → c, c ∈ IR in Theorem 5.86 was replaced
P
with Yn → c, c ∈ IR. This by no means a contradiction because these two conditions
are equivalent according to Proposition 5.70.
•
247
Exercise 5.88 — Slutsky’s theorem or preservation of d−convergence under
(restricted) addition
Prove Theorem 5.86 (Karr, 1993, p. 146; Rohatgi, 1976, pp. 253–4).
•
As for the product, almost sure convergence and convergence in probability are
preserved.
Theorem 5.89 — Preservation of {a.s., P }−convergence under product (Karr,
1993, p. 147)
∗
∗
Let Xn → X and Yn → Y , where ∗ = a.s., P . Then
∗
Xn × Yn → X × Y, ∗ = a.s., P.
(5.43)
•
Exercise 5.90 — Preservation of {a.s., P }−convergence under product
Show Theorem 5.89 (Karr, 1993, p. 147).
•
Theorem 5.91 — (Non)preservation of q.m.−convergence under product (Karr,
1993, p. 147)
q.m.
q.m.
Let Xn → X and Yn → Y . Then
L1
Xn × Yn → X × Y.
(5.44)
•
Remark 5.92 — (Non)preservation of q.m.−convergence under product (Karr,
1993, pp. 146–7)
Quadratic mean convergence of products does not hold in general, since X × Y need not
belong to L2 when X and Y do:
q.m.
q.m.
Xn → X, Yn → Y
q.m.
6⇒ Xn × Yn → X × Y.
(5.45)
However, the product of r.v. in L2 belongs to L1 , and L2 convergence of factors implies
L1 convergence of products.
•
Exercise 5.93 — (Non)preservation of q.m.−convergence under product
Prove Theorem 5.91 (Karr, 1993, p. 147; Rohatgi, 1976, p. 254).
248
•
Convergence in distribution is preserved under product, provided that one limit factor
is constant (Karr, 1993, p. 146).
Theorem 5.94 — Slutsky’s theorem (bis) or preservation of d−convergence
under (restricted) product (Karr, 1993, p. 147)
Let:
d
• Xn → X;
d
• Yn → c, c ∈ IR.
Then
d
Xn × Yn → X × c.
(5.46)
•
Remark 5.95 — Slutsky’s theorem or preservation of d−convergence under
(restricted) product (Rohatgi, 1976, p. 253)
Theorem 15 (Rohatgi, 1976, p. 253) also states that
d
P
d
Xn → X, Yn → c, c ∈ IR ⇒ Xn × Yn → X × c
Xn d X
d
P
→ .
Xn → X, Yn → c, c ∈ IR\{0} ⇒
Yn
c
(5.47)
(5.48)
•
(Discuss the validity of both results.)
Preservation under...
Mode of convergence
a.s.
→ (almost sure)
P
→ (in probability)
q.m.
→ (in quadratic mean)
L1
→ (in L1 )
d
→ (in distribution)
∗
Continuous mapping
Addition & Subtraction
Product
Yes
Yes
Yes
Yes
Yes
Yes
No
Yes
→
No
Yes
Yes
RV∗
RV∗
Yes
L1
Restricted validity (RV): one of the summands/factors has to converge in distribution to a constant
Exercise 5.96 — Slutsky’s theorem or preservation of d−convergence under
(restricted) product
Prove Theorem 5.94 (Karr, 1993, pp. 147–8).
•
249
Example/Exercise 5.97 — Slutsky’s theorem or preservation of d−convergence
under (restricted) product
P
i.i.d.
Consider the sequence of r.v. {X1 , X2 , . . .}, where Xn ∼ X and let X̄n = n1 ni=1 Xi and
Pn
1
2
Sn2 = n−1
i=1 (Xi − X̄n ) be the sample mean and the variance of the first n r.v.
(a) Show that
X̄n − µ d
√ → Normal(0, 1),
Sn / n
(5.49)
for any X ∈ L4 .
• R.v.
i.i.d.
Xi ∼ X, i ∈ IN
X : E(X) = µ, V (X) = σ 2 = µ2 , E [(X − µ)4 ] = µ4 , which are finite
moments since X ∈ L4 .
• Auxiliary results
E(X̄n ) = µ
V (X̄n ) =
σ2
n
2
µ2
n
=
E(Sn2 ) = σ = µ2
2 h µ4 −µ2
n
2
V (Sn2 ) = n−1
−
n
2(µ4 −2µ22 )
n2
+
• Asymptotic sample distribution of
To show that
2(µ4 −3µ22 )
n3
i
(Murteira, 1980, p. 46).
X̄n −µ
√
Sn / n
d
X̄n −µ
√
Sn / n
→ Normal(0, 1) it suffices to note that
X̄n −µ
√
X̄n − µ
σ/ n
√ = q ,
2
Sn / n
Sn
(5.50)
σ2
X̄n√
−µ
σ/ n
d
prove that
→ Normal(0, 1) and
theorem as stated in (5.48).
q
2
Sn
σ2
P
→ 1, and then apply Slutsky’s
• Convergence in distribution of the numerator
It follows immediately from the Central Limit Theorem.5
5
This well known theorem is thoroughly discussed by Karr (1993, pp. 190–196).
250
• Convergence in probability of the denominator
By using the definition of convergence in probability and the Chebyshev(Bienaymé)’s inequality, we get, for any > 0:
!
p
V (Sn2 )
lim P |Sn2 − σ 2 | > = lim P Sn2 − E(Sn2 ) ≥ p
n→+∞
n→+∞
V (Sn2 )
1
≤ lim 2
n→+∞
√ 2
V (Sn )
1
lim V (Sn2 )
2 n→+∞
= 0,
=
(5.51)
P
i.e. Sn2 → σ 2 .
Finally, note that convergence in probability is preserved under continuous
p
mappings such as g(x) = σx , hence
r
r
Sn2 P
σ2
2 P
2
Sn → σ ⇒
→
= 1.
(5.52)
σ2
σ2
• Conclusion
X̄−µ
√
S/ n
d
→ Normal(0, 1).
•
(b) Discuss the utility of this result.
Exercise 5.98 — Slutsky’s theorem or preservation of d−convergence under
(restricted) division
i.i.d.
Let Xi ∼ Normal(0, 1), i ∈ IN .
Determine the limiting distribution of Wn = UVnn , where
Pn
i=1 Xi
√
Un =
(5.53)
n
Pn
2
i=1 Xi
Vn =
,
(5.54)
n
by proving that
d
Un → Normal(0, 1)
(5.55)
d
Vn → 1
(5.56)
•
(Rohatgi, 1976, pp. 254, Example 12).
251
Exercise 5.99 — Slutsky’s theorem or preservation of d−convergence under
(restricted) division (bis)
Let {X1 , X2 , . . .} a sequence of i.i.d. r.v. with common distribution Bernoulli(p) and X̄n =
Pn
1
i=1 Xi the maximum likelihood estimator of p.
n
(a) Prove that
X̄ − p
q n
d
→ Normal(0, 1).
(5.57)
X̄n (1−X̄n )
n
(b) Discuss the relevance of this convergence in distribution.
252
•
5.4
Convergence of random vectors
Before defining modes of convergence of a sequence of random d−vectors we need two
recall the definition of norm of a vector.
Definition 5.100 — L2 (or Euclidean) and L1 norms of x (Karr, 1993, p. 149)
Let x ∈ IRd and x(i) its ith component. Then
v
u d
uX
||x||L2 = t
(5.58)
x(i)2
i=1
||x||L1
v
u d
uX
= t
|x(i)|
(5.59)
i=1
2
denote the L norm (or Euclidean norm) and the L1 norm of x, respectively.
•
Remark 5.101 — L2 (or Euclidean) and L1 norms of x
(http://en.wikipedia.org/wiki/Norm mathematics]Definition)
On IRd , the intuitive notion of length of the vector x is captured by its L2 or Euclidean
norm: this gives the ordinary distance from the origin to the point x, a consequence of
the Pythagorean theorem.
The Euclidean norm is by far the most commonly used norm on IRd , but there are
other norms, such as the L1 norm on this vector space.
•
Definition 5.102 — Four modes of convergence (as functions of Ω) of sequences
of random vectors (Karr, 1993, p. 149)
∗
Let X, X 1 , X 2 , . . . be random d−vectors. Then the four modes of convergence X n → X,
∗ = a.s., P, q.m., L1 are natural extensions of their counterparts in the univariate case:
a.s.
• X n → X if P ({ω : limn→+∞ ||Xn (ω) − X(ω)||L1 }) = 1;
P
• X n → X if limn→+∞ P ({||X n − X||L1 > }) = 0, for every > 0;
q.m.
• X n → X if limn→+∞ E (||X n − X||L2 ) = 0;
L1
• X n → X if limn→+∞ E (||X n − X||L1 ) = 0.
•
Proposition 5.103 — Alternative criteria for the four modes of convergence of
sequences of random vectors (Karr, 1993, p. 149)
∗
X n → X, ∗ = a.s., P, q.m., L1 iff the same kind of stochastic convergence holds for each
∗
component, i.e. X n (i) → X(i), ∗ = a.s., P, q.m., L1 .
•
253
Remark 5.104 — Convergence in distribution of a sequence of random vectors
(Karr, 1993, p. 149)
Due to the intactability of multi-dimension d.f., convergence in distribution — unlike
the four previous modes of convergence — has to be defined by taking advantage of the
alternative criterion for convergence in distribution stated in Theorem 5.49.
•
Definition 5.105 — Convergence in distribution of a sequence of random
vectors
(Karr, 1993, p. 149)
Let X, X 1 , X 2 , . . . be random d−vectors. Then:
d
• X n → X if E[f (X n )] → E[f (X)], for all bounded, continuous functions f : IRd →
IR.
•
Proposition 5.106 — A sufficient condition for the convergence in distribution
of the components of a sequence of random vectors (Karr, 1993, p. 149)
Unlike the four previous modes of convergence, convergence in distribution of the
components of a sequence of random vectors is implied, but need not imply, convergence
in distribution of the sequence of random vectors:
d
d
X n → X ⇒ (6⇐) X n (i) → X(i),
(5.60)
•
for each i.
A sequence of random vectors converges in distribution iff every linear combination
of their components converges in distribution; this result constitutes the Cramér-Wold
device.
Theorem 5.107 (Cramér-Wold device) — An alternative criterion for the
convergence in distribution of a sequence of random vectors (Karr, 1993, p.
150)
Let X, X 1 , X 2 , . . . be random d−vectors. Then
d
>
Xn → X ⇔ a Xn =
d
X
d
a(i) × X n (i) →
i=1
d
X
a(i) × X(i) = a> X,
(5.61)
i=1
for all a ∈ IRd .
•
Exercise 5.108 — An alternative criterion for the convergence in distribution
of a sequence of random vectors
Show Theorem 5.107 (Karr, 1993, p. 150).
•
254
As with sequences of r.v., convergence almost surely, in probability and in distribution
are preserved under continuous mappings of sequences of random vectors.
Theorem 5.109 — Preservation of {a.s., P, d}−convergence under continuous
mappings of random vectors (Karr, 1993, p. 148)
Let:
• {X 1 , X 2 , . . .} be a sequence of random d−vectors and X a random m−vector;
• g : IRd → IRm be a continuous mapping of IRd into IRm .
Then
∗
Xn → X
∗
⇒ g(X n ) → g(X), ∗ = a.s., P, d.
(5.62)
•
255
5.5
Limit theorems for Bernoulli summands
Let {X1 , X2 , . . .} be a Bernoulli process with parameter p ∈ (0, 1). In this section we
study the asymptotic behavior of the Bernoulli counting process {S1 , S2 , . . .}, where Sn =
Pn
i=1 Xi ∼ Binomial(n, p).
5.5.1
Laws of large numbers for Bernoulli summands
Motivation 5.110 — Laws of large numbers
(http://en.wikipedia.org/wiki/Law of large numbers; Murteira, 1979, p. 313)
In probability theory, the law of large numbers (LLN) is a theorem that describes the
result of performing the same experiment a large number of times. According to the law,
the average of the results obtained from a large number of trials (e.g. Bernoulli trials)
should be close to the expected value, and will tend to become closer as more trials are
performed.
For instance, when a fair coin is flipped once, the expected value of the number of
heads is equal to one half. Therefore, according to the law of large numbers, the proportion
of heads in a large number of coin flips should be roughly one half, as depicted by the
next figure (where N stands for n).
This illustration suggests the following statement: Snn = X̄n converges, in some sense, to
p = 21 . In fact, if we use Chebyshev(-Bienaymé)’s inequality we can prove that
Sn
lim P
−p >ε
= 0,
(5.63)
n→+∞
n
P
that is, Snn → p = 12 . (Show this result!) In addition, we can also prove that the proportion
of heads after n flips will almost surely converge to one half as n approaches infinity, i.e.,
Sn a.s.
→ p = 12 . Similar convergences can be devised for the mean of n i.i.d. r.v.
n
The Indian mathematician Brahmagupta (598–668) and later the Italian
mathematician Gerolamo Cardano (1501–1576) stated without proof that the accuracies
of empirical statistics tend to improve with the number of trials. This was then formalized
as a law of large numbers (LLN).
256
The LLN was first proved by Jacob Bernoulli. It took him over 20 years to develop
a sufficiently rigorous mathematical proof which was published in his Ars Conjectandi
(The Art of Conjecturing) in 1713. He named this his ”Golden Theorem” but it became
generally known as ”Bernoulli’s Theorem”. In 1835, S.D. Poisson further described it
under the name ”La loi des grands nombres” (”The law of large numbers”). Thereafter,
it was known under both names, but the ”Law of large numbers” is most frequently used.
Other mathematicians also contributed to refinement of the law, including Chebyshev,
Markov, Borel, Cantelli and Kolmogorov. These further studies have given rise to two
prominent forms of the LLN:
• the weak law of large numbers (WLLN);
• the strong law of large numbers (SLLN);
These forms do not describe different laws but instead refer to different ways of describing
the mode of convergence of the cumulative sample means to the expected value:
• the WLLN refers to a convergence in probability;
• the SLLN is concerned with an almost sure convergence;
Needless to say that the SLLN implies the WLLN.
•
Theorem 5.111 — Weak law of large numbers for Bernoulli summands
(Karr, 1993, p. 151)
Let:
• {X1 , X2 , . . .} be a Bernoulli process with parameter p ∈ (0, 1);
•
Sn
n
= X̄n be the proportion of successes in the first n Bernoulli trials.
Then
Sn q.m.
→ p,
n
(5.64)
therefore
Sn P
→ p.
n
(5.65)
•
257
Exercise 5.112 — Weak law of large numbers for hBernoulliisummands
2
(thus proving the
Show Theorem 5.111, by calculating the limit of E Snn − p
convergence in quadratic mean) and combining Proposition 5.55 (which states that
convergence in quadratic mean implies convergence in L1 ) and Proposition 5.57 (it says
that convergence in L1 implies convergence in probability) (Karr, 1993, p. 151).
•
Theorem 5.113 — Strong law of large numbers for Bernoulli summands or
Borel’s SLLN (Karr, 1993, p. 151; Rohatgi, 1976, p. 273, Corollary 3)
Let:
• {X1 , X2 , . . .} be a Bernoulli process with parameter p ∈ (0, 1);
•
Sn
n
= X̄n be the proportion of successes in the first n Bernoulli trials.
Then
Sn a.s.
→ p.
n
(5.66)
•
Exercise 5.114 — Strong law of large numbers for Bernoulli summands or
Borel’s SLLN
Prove Theorem 5.113, by: using Theorem 4.121 (Chebyshev’s inequality) with g(x) = x4
to set an upper limit to P ({|Sn − np| > n}), which is smaller than O(n−2 ),6 thus, proving
Sn
P+∞
| n − p| > that
< ∞, i.e., that the sequence { S11 , S22 , . . .} completely
i=1 P
converges to p; finally applying Proposition 5.44 which relates almost sure convergence
and complete convergence (Karr, 1993, pp. 151–152).
•
Remark 5.115 — Weak and strong laws of large numbers for Bernoulli
summands (http://en.wikipedia.org/wiki/Law of large numbers; Karr, 1993, p. 152)
• Theorem 5.113 can be invoked to support the frequency interpretation of probability.
• The WLLN for Bernoulli summands states that for a specified large n, Snn is likely
to be near p. Thus, it leaves open the possibility that the event {| Snn − p| > }, for
any > 0, happens an infinite number of times, although at infrequent intervals.
The SLLN for Bernoulli summands shows that this almost surely will not occur.
In particular, it implies that with probability 1, we have that, for any > 0, the
inequality | Snn − p| > holds for all large enough n.
6
Let f (x) and g(x) be two functions defined on some subset of the real numbers. One writes f (x) =
O(g(x)) as x → ∞ iff there exists a positive real number M and a real number x0 such that |f (x)| ≤
M |g(x)| for all x > x0 (http://en.wikipedia.org/wiki/Big O notation).
258
• Finally, the proofs of theorems 5.111 and 5.113 only involve the moments of Xi .
Unsurprisingly, these two theorems can be reproduced for other sequences of i.i.d.
r.v., namely those in L2 (in the case of the WLLN) and in L4 (for the SLLN), as we
shall see in sections 5.6 and 5.7.
•
5.5.2
Central limit theorems for Bernoulli summands
Motivation 5.116 — Central limit theorems for Bernoulli summands (Karr,
1993, p. 152)
They essentially state that, in the Bernoulli summands case and for large n,
n −E(Sn )
= √Sn −np has approximately a standard
Sn ∼ Binomial(n, p) is such that S√
V (Sn )
np(1−p)
normal distribution.
The local (resp. global) central limit theorem — also known as the DeMoivre-Laplace
local (resp. global) limit theorem — provides an approximation to the p.f. (resp. d.f.) of
Sn in terms of the standard normal p.d.f. (resp. d.f.).
•
Theorem 5.117 — DeMoivre-Laplace local limit theorem (Karr, 1993, p. 153)
Let:
• kn = 0, 1, . . . , n;
• xn = √kn −np
np(1−p)
• φ(x) =
= o(n1/6 );7
2
√1 e−x /2
2π
be the standard normal p.d.f.
Then
P ({Sn = kn })
= 1.
n→+∞
√φ(xn )
(5.67)
lim
np(1−p)
•
Exercise 5.118 — DeMoivre-Laplace local limit theorem
Show Theorem 5.117 (Karr, 1993, p. 153).
7
The relation f (x) = o(g(x)) is read as “f (x) is little-o of g(x)”.
Intuitively,
(x)
means that g(x) grows much faster than f (x).
Formally, it states limx→∞ fg(x)
=
(http://en.wikipedia.org/wiki/Big O notation]Little-o notation).
259
•
it
0
Remark 5.119 — DeMoivre-Laplace local limit theorem (Karr, 1993, p. 153)
The proof of Theorem 5.117 shows that the convergence in (5.67) is uniform in values of
k satisfying |k − np| = o(n2/3 ). As a consequence, for large values of n and values of kn
not to different from np,
"
#
kn − np
1
×φ p
,
(5.68)
P ({Sn = kn }) ' p
np(1 − p)
np(1 − p)
that is, the p.f. of Sn ∼ Binomial(n, p) evaluated at kn can be properly approximated by
the p.d.f. of a normal distribution, with mean E(Sn ) = np and variance V (Sn ) = np(1−p),
•
evaluated at √kn −np .
np(1−p)
Theorem 5.120 — DeMoivre-Laplace global limit theorem (Karr, 1993, p. 154;
Murteira, 1979, p. 347)
Let Sn ∼ Binomial(n, p), n ∈ IN . Then
S − np
d
p n
→ Normal(0, 1).
np(1 − p)
(5.69)
•
Remark 5.121 — DeMoivre-Laplace global limit theorem (Karr, 1993, p. 155;
Murteira, 1979, p. 347)
• Theorem 5.120 justifies the following approximation:
"
x − np
#
P (Sn ≤ x) ' Φ p
.
np(1 − p)
(5.70)
• According to Murteira (1979, p. 348), the well known continuity correction was
proposed by Feller in 1968 to improve the normal approximation to the binomial
distribution,8 and can be written as:
"
#
"
#
a − 12 − np
b + 12 − np
P (a ≤ Sn ≤ b) ' Φ p
−Φ p
.
np(1 − p)
np(1 − p)
8
(5.71)
However, http://en.wikipedia.org/wiki/Continuity correction suggests that continuity correction
dates back from Feller, W. (1945). On the normal approximation to the binomial distribution. The
Annals of Mathematical Statistics 16, pp. 319–329.
260
• The proof of the central limit theorem for summands (other than Bernoulli
ones) involves a Taylor series expansion9 and requires dealing with the notion of
characteristic function of a r.v.10 Such proof can be found in Murteira (1979, pp.
354–355); Karr (1993, pp. 190–196) devotes a whole section to this theorem.
•
Exercise 5.122 — DeMoivre-Laplace global limit theorem
Show Theorem 5.120 (Karr, 1993, pp. 154–155).
5.5.3
•
The Poisson limit theorem
Motivation 5.123 — Poisson limit theorem
(Karr, 1993, p. 155; http://en.wikipedia.org/wiki/Poisson limit theorem)
• In the two central limit theorems for Bernoulli summands, although n → +∞,
the parameter p remained fixed. These theorems provide useful approximations to
binomial probabilities, as long as the values of p are close to neither zero or one,
and inaccurate ones, otherwise.
• The Poisson limit theorem gives a Poisson approximation to the binomial
distribution, under certain conditions, namely, it considers the effect of
simultaneously allowing n → +∞ and p = pn → 0 with the proviso that n×pn → λ,
where λ ∈ IR+ . This theorem was obviously named after Siméon-Denis Poisson
(1781–1840).
•
Theorem 5.124 — Poisson limit theorem (Karr, 1993, p. 155)
Let:
• {Y1 , Y2 , . . .} be a sequence of r.v. such that Yn ∼ Binomial(n, pn ), for each n;
• n × pn → λ, where λ ∈ IR+ .
Then
d
Yn → Poisson(λ).
(5.72)
•
9
The Taylor series of a real or complex function f (x) that is infinitely differentiable in a neighborhood
of a real (or complex number) a is the power series written in the more compact sigma notation as
P+∞ f (n) (a)
(x − a)n , where f (n) (a) denotes the nth derivative of f evaluated at the point a. In the case
n=0
n!
that a = 0, the series is also called a Maclaurin series (http://en.wikipedia.org/wiki/Taylor series).
10
For a scalar random variable X the characteristic function is defined as the expected value of eitX ,
E(eitX ), where i is the imaginary unit, and t ∈ IR is the argument of the characteristic function
(http://en.wikipedia.org/wiki/Characteristic function (probability theory)).
261
Example/Exercise 5.125 — Poisson limit theorem
(a) Consider 0 < λ < n and let us verify that
λx
n x
lim
pn (1 − pn )n−x = e−λ .
n → +∞
x
x!
pn → 0
npn = λ fix
• R.v.
Xn ∼ Binomial(n, pn )
• Parameters
n
pn =
λ
n
(0 < λ < n)
• P.f.
P (Xn = x) =
n
x
λ x
n
1−
λ n−x
n
, x = 0, 1, . . . , n
• Limit p.f.
For any x ∈ {0, 1, . . . , n}, we get
λx
n(n − 1) . . . (n − x + 1)
× lim
x! n→+∞
nx
n
−x
−λ
λ
× lim 1 +
× lim 1 −
n→+∞
n→+∞
n
n
x
λ
× 1 × e−λ × 1
=
x!
λx
= e−λ .
x!
lim P (Xn = x) =
n→+∞
• Conclusion
If the limit p.f. of Xn coincides with p.f. of X ∼ Poisson(λ) then the same holds
for the limit d.f. of Xn and the d.f. of X. Hence
d
Xn → Poisson(λ).
(b) Now, prove Theorem 5.124 (Karr, 1993, p. 155).
(c) Suppose that in an interval of length 1000, 500 points are placed randomly.
Use the Poisson limit theorem to prove that we can approximate the p.f. of the number
points that will be placed in a sub-interval of length 10 by
5k
k!
(http://en.wikipedia.org/wiki/Poisson limit theorem).
e−5
262
(5.73)
•
5.6
Weak law of large numbers
Motivation 5.126 — Weak law of large numbers (Rohatgi, 1976, p. 257)
Let:
• {X1 , X2 , . . .} be a sequence of r.v. in L2 ;
• Sn =
Pn
i=1
Xi be the sum of the first n terms of such a sequence.
In this section we are going to answer the next question in the affirmative:
• Are there constants an and bn (bn > 0) such that
Sn −an P
→
bn
0?
In other words, what follows are extensions of the WLLN for Bernoulli summands
(Theorem 5.111), to other sequences of:
• i.i.d. r.v. in L2 ;
• pairwise uncorrelated and identically distributed r.v. in L2 ;
• pairwise uncorrelated r.v. in L2 ;
• r.v. in L2 with a specific variance behavior;
• i.i.d. r.v. in L1 .
•
Definition 5.127 — Obeying the weak law of large numbers (Rohatgi, 1976, p.
257)
Let:
• {X1 , X2 , . . .} be a sequence of r.v.;
• Sn =
Pn
i=1
Xi , n = 1, 2, . . .;
Then {X1 , X2 , . . .} is said to obey the weak law of large numbers (WLLN) with respect
to the sequence of constants {b1 , b2 , . . .} (bn > 0, bn ↑ +∞) if there is a sequence of real
constants {a1 , a2 , . . .} such that
Sn − an P
→ 0.
bn
(5.74)
an and bn are called centering and norming constants, respectively.
263
•
Remark 5.128 — Obeying the weak law of large numbers (Rohatgi, 1976, p. 257)
The definition in Murteira (1979, p. 319) is a particular case of Definition 5.127 with
P
an = ni=1 E(Xi ) and bn = n.
P
• Let {X1 , X2 , . . .} be a sequence of r.v., X̄n = n1 ni=1 Xi , and {Z1 , Z2 , . . .} be another
n
sequence of r.v. such that Zn = Snb−a
= X̄n − E(X̄n ), n = 1, 2, . . .
n
P
Then {X1 , X2 , . . .} is said to obey the WLLN if Zn → 0.
Hereafter the convergence results are stated either in terms of Sn or X̄n .
•
Theorem 5.129 — Weak law of large numbers, i.i.d. r.v. in L2 (Karr, 1993, p.
152)
Let {X1 , X2 , . . .} be a sequence of i.i.d. r.v. in L2 with common expected value µ and
variance σ 2 . Then
q.m.
X̄n → µ,
(5.75)
therefore {X1 , X2 , . . .} obeys the WLLN:
P
X̄n → µ,
i.e.,
Sn −nµ P
→
n
(5.76)
0.11
•
Exercise 5.130 — Weak law of large numbers, i.i.d. r.v. in L2
Prove Theorem 5.129, by mimicking the proof of the WLLN for Bernoulli summands. •
Exercise 5.131 — Weak law of large numbers, i.i.d. r.v. in L2 (bis)
Let {X1 , X2 , . . .} a sequence of i.i.d. r.v. with common pd.f.
(
e−x+q , x > q
f (x) =
0,
otherwise
(a) Prove that X̄n =
1
n
Pn
i=1
(5.77)
P
Xi → 1 + q.
P
(b) Show that X(1:n) = mini=1,...,n Xi → q.12
•
A closer look to the proof Theorem 5.129 leads to the conclusion that the r.v. need
only be pairwise uncorrelated and identically distributed in L2 , since in this case we still
2
have V (X̄n ) = σn (Karr, 1993, p. 152).
11
12
As suggested by Rohatgi (1976, p. 258, Corollary 3).
Use the d.f. of X(1:n) .
264
Theorem 5.132 — Weak law of large numbers, pairwise uncorrelated and
identically distributed r.v. in L2 (Karr, 1993, p. 152; Rohatgi, 1976, p. 258)
Let {X1 , X2 , . . .} be a sequence of pairwise uncorrelated and identically distributed r.v.
q.m.
in L2 with common expected value µ and variance σ 2 . Thus, X̄n → µ and13 hence
{X1 , X2 , . . .} obeys the WLLN:
P
X̄n → µ.
(5.78)
•
We can also drop the assumption that we are dealing with identically distributed r.v.
as suggested by the following theorem.
Theorem 5.133 — Weak law of large numbers, pairwise uncorrelated r.v. in
L2 (Rohatgi, 1976, p. 258, Theorem 1)
Let
• {X1 , X2 , . . .} be a sequence of pairwise uncorrelated r.v. in L2 with E(Xi ) = µi and
V (Xi ) = σi2 ;
P
• an = ni=1 µi ;
P
• bn = ni=1 σi2 .
If bn → +∞ then
P
Pn
Xi − ni=1 µi P
Sn − an
i=1 P
→ 0,
=
n
2
bn
i=1 σi
(5.79)
i.e., {X1 , X2 , . . .} obeys the WLLN with respect to bn .
•
Exercise 5.134 — Weak law of large numbers, pairwise uncorrelated r.v. in L2
Prove Theorem 5.133 by applying Chebyshev(-Bienaymé)s inequality (Rohatgi, 1976, p.
258).
•
Remark 5.135 — Weak law of large numbers, pairwise uncorrelated r.v. in L2
A careful inspection of the proof of Theorem 5.133 (Rohatgi, 1976, p. 258) leads us to
restate it as follows:
• Let {X1 , X2 , . . .} be a sequence of pairwise uncorrelated r.v. in L2 with E(Xi ) = µi
P
and V (Xi ) = σi2 , an = ni=1 µi , and
13
Rohatgi (1976, p. 258, Corollary 1) does not refer this convergence in quadratic mean.
265
v
u n
uX
bn = t
σi2 .
(5.80)
i=1
If bn → +∞ then
Sn − E(Sn ) P
Sn − an
= p
→ 0.
bn
V (Sn )
(5.81)
Theorem 5.133 can be further generalized: the sequence of r.v. need only have the mean
of its first n terms, X̄n , with a specific variance behavior, as stated below.
•
Theorem 5.136 — WLLN and Markov’s theorem (Murteira, 1979, p. 320)
Let {X1 , X2 , . . .} be a sequence of r.v. in L2 . If
!
n
X
1
(5.82)
lim V (X̄n ) = lim 2 V
Xi = 0,
n→+∞
n→+∞ n
i=1
then
P
X̄n − E(X̄n ) → 0,
(5.83)
that is, {X1 , X2 , . . .} obeys the WLLN with respect to bn = n (an =
Pn
i=1
E(Xi )).
Exercise 5.137 — WLLN and Markov’s theorem
Show Theorem 5.136, by simply applying Chebyshev(-Bienaymé)’s inequality.
•
•
Remark 5.138 — (Special cases of ) Markov’s theorem (Murteira, 1979, pp. 320–
321; Rohatgi, 1979, p. 258)
• The WLLN holds for a sequence of pairwise uncorrelated r.v., with common expected
value µ and uniformly limited variance V (Xn ) < k, n = 1, 2, . . . ; k ∈ IR+ .14
• The WLLN also holds for a sequence of pairwise uncorrelated and identically
distributed r.v. in L2 , with common expected value µ and σ 2 (Theorem 5.132).
14
This corollary of Markov’s theorem is due to Chebyshev. Please note that when we dealing with
pairwise uncorrelated r.v., the condition (5.82) in Markov’s theorem still reads: limn→+∞ V (X̄n ) =
Pn
limn→+∞ n12 i=1 V (Xi ) = 0.
266
• Needless to say that the WLLN holds for any a sequence of i.i.d. r.v. in L2 (Theorem
5.129) and therefore X̄n is a consistent estimator of µ.
Moreover, according to http://en.wikipedia.org/wiki/Law of large numbers, the
assumption of finite variances (V (Xi ) = σ 2 < +∞) is not necessary; large or infinite
variance will make the convergence slower, but the WLLN holds anyway, as stated
in Theorem 5.143. This assumption is often used because it makes the proofs easier
and shorter.
•
The next theorem provides a necessary and sufficient condition for a sequence of r.v.
{X1 , X2 , . . .} to obey the WLLN.
Theorem 5.139 — An alternative criterion for the WLLN (Rohatgi, 1976, p. 258)
Let:
• {X1 , X2 , . . .} be a sequence of r.v. (in L2 );
• Ȳn = X̄n − E(X̄n ), n = 1, 2, . . ..
Then {X1 , X2 , . . .} satisfies the WLLN with respect to bn = n (an =
P
X̄n − E(X̄n ) → 0,
Pn
i=1
E(Xi )), i.e.
(5.84)
iff
lim E
n→+∞
Yn2
1 + Yn2
= 0.
(5.85)
•
Remark 5.140 — An alternative criterion for the WLLN (Rohatgi, 1976, p. 259)
Since condition (5.85) does not apply to the individual r.v. Xi Theorem 5.139 is of limited
use.
•
Exercise 5.141 — An alternative criterion for the WLLN
Show Theorem 5.139 (Rohatgi, 1976, pp. 258–259).
•
Exercise 5.142 — An alternative criterion for the WLLN (bis)
Let (X1 , . . . , Xn ) be jointly normal and such that: E(Xi ) = 0 and V (Xi ) = 1 (i = 1, 2, . . .);
and,


i=j
 1,
cov(Xi , Xj ) =
(5.86)
ρ ∈ (−1, 1), |j − i| = 1


0,
|j − i| > 1.
P
Use Theorem 5.139 to prove that X̄n → 0 (Rohatgi, 1976, pp. 250–260, Example 2).
267
•
Finally, the assumption that the r.v. belong to L2 is dropped and we state a theorem
due to Soviet mathematician Aleksandr Yakovlevich Khinchin (1894–1959).
Theorem 5.143 — Weak law of large numbers, i.i.d. r.v. in L1 (Rohatgi, 1976, p.
261)
Let {X1 , X2 , . . .} be a sequence of i.i.d. r.v. in L1 with common finite mean µ;15 Then
{X1 , X2 , . . .} satisfies the WLLN with respect to bn = n (an = nµ), i.e.
P
X̄n → µ.
(5.87)
•
Exercise 5.144 — Weak law of large numbers, i.i.d. r.v. in L1
Prove Theorem 5.143 (Rohatgi, 1976, p. 261).
•
Exercise 5.145 — Weak law of large numbers, i.i.d. r.v. in L1 (bis)
Let {X1 , X2 , . . .} be a sequence of i.i.d. r.v. to X ∈ Lk , for some positive integer k, and
common kth. moment E(X k ). Apply Theorem 5.143 to prove that:
(a)
1
n
Pn
i=1
P
Xik → E(X k );16
(b) if k = 2 then
1
n
Pn
i=1
P
Xi2 − (X̄n )2 → V (X)
17
(Rohatgi, 1976, p. 261, Example 4). •
Exercise 5.146 — Weak law of large numbers, i.i.d. r.v. in L1 (bis bis)
Let {X1 , X2 , . . .} be a sequence of i.i.d. r.v. with common p.d.f.
(
1+δ
, x≥1
x2+δ
fX (x) =
0,
otherwise,
P
where δ > 0.18 Show that X̄n → E(X) =
1+δ
δ
(Rohatgi, 1976, p. 261, Example 5).
15
(5.88)
•
Please note that nothing is said about the variance, it need not to be finite.
Pn
This means that the kth. sample moment, n1 i=1 Xik , is a consistent estimator of E(X k ) if the i.i.d.
r.v. belong to Lk .
Pn
17
I.e., the sample variance, n1 i=1 Xi2 − (X̄n )2 , is a consistent estimator of V (X) if we are dealing
with i.i.d. r.v. in L2 .
18
This is the p.d.f. of a Pareto(1, 1 + δ) r.v.
16
268
5.7
Strong law of large numbers
This section is devoted to a few extensions of the SLNN for Bernoulli summands (or
Borel’s SLLN), Theorem 5.113. They refer to sequences of:
• i.i.d. r.v. in L4 ;
• dominated i.i.d. r.v.;
• independent r.v. in L2 with a specific variance behavior;
• i.i.d. r.v. in L1 .
Theorem 5.147 — Strong law of large numbers, i.i.d. r.v. in L4 (Karr, 1993, p.
152; Rohatgi, 1976, pp. 264–265, Theorem 1)
Let {X1 , X2 , . . .} be a sequence of i.i.d. r.v. in L4 , with common expected value µ. Then
a.s.
X̄n → µ.
(5.89)
•
Exercise 5.148 — Strong law of large numbers, i.i.d. r.v. in L4
Prove Theorem 5.147, by following the same steps as in the proof of the SLLN for Bernoulli
summands (Rohatgi, 1976, p. 265).
•
The proviso of a common finite fourth moment can be dropped if there is a degenerate
r.v. that dominates the i.i.d. r.v. X1 , X2 , . . .
Corollary 5.149 — Strong law of large numbers, dominated i.i.d. r.v. (Rohatgi,
1976, p. 265) Let {X1 , X2 , . . .} be a sequence of i.i.d. r.v., with common expected value
µ and such that
P ({|Xn | < k}) , for all n,
(5.90)
where k is a positive constant. Then
a.s.
X̄n → µ.
(5.91)
•
The next lemma is essential to prove yet another extension of Borel’s SLLN.
269
Lemma 5.150 — Kolmogorov’s inequality (Rohatgi, 1976, p. 268)
Let {X1 , X2 , . . .} be a sequence of independent r.v., with common null mean and variances
σi2 , i = 1, 2, . . . Then, for any > 0,
Pn
σi2
P
max |Sk | > ≤ i=1
.
(5.92)
k=1,...,n
2
•
Exercise 5.151 — Kolmogorov’s inequality
Show Lemma 5.150 (Rohatgi, 1976, pp. 268–269).
Remark 5.152 — Kolmogorov’s inequality (Rohatgi, 1976, p. 269)
If we take n = 1 then Lemma 5.150 can be written as P ({|X1 | > }) ≤
Chebyshev’s inequality.
•
σ12
,
2
which is
•
The condition of dealing with i.i.d. r.v. in L4 can be further relaxed as long as the r.v.
are still independent and the variances of X1 , X2 , . . . have a specific behavior, as stated
below.
Theorem 5.153 — Strong law of large numbers, independent r.v. in L2 (Rohatgi,
1976, p. 272, Theorem 6)
Let {X1 , X2 , . . .} be a sequence of independent r.v. in L2 with variances σi2 , i = 1, 2, . . .,
such that
+∞
X
V (Xi ) < +∞.
(5.93)
i=1
Then
a.s.
Sn − E(Sn ) → 0.
(5.94)
•
Exercise 5.154 — Strong law of large numbers, independent r.v. in L2
Prove Theorem 5.153 by making use of Kolmogorov’s inequality and Cauchy’s criterion
(Rohatgi, 1976, p. 272, Theorem 6)
•
Theorem 5.155 — Strong law of large numbers, i.i.d. r.v. in L1 ,
Kolmogorov’s SLLN (Karr, 1993, p. 188; Rohatgi, 1976, p. 274, Theorem 7)
Let {X1 , X2 , . . .} be a sequence of i.i.d. r.v. to X. Then
a.s.
X̄n → µ
or
(5.95)
iff X ∈ L1 , and then µ = E(X).
•
270
Exercise 5.156 — Strong law of large numbers, i.i.d. r.v. in L1 ,
Kolmogorov’s SLLN
Show Theorem 5.155 (Karr, 1993, pp. 188–189; Rohatgi, 1976, pp. 274–275)
or
•
References
• Grimmett, G.R. and Stirzaker, D.R. (2001). Probability and Random Processes
(3rd. edition). Oxford University Press. (QA274.12-.76.GRI.30385 and QA274.12.76.GRI.40695 refer to the library code of the 1st. and 2nd. editions from 1982 and
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